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\begin{document}
\frontmatter
\title{Height, graded relative hyperbolicity and quasiconvexity}
\author[\initial{F.} \lastname{Dahmani}]{\firstname{François} \lastname{Dahmani}}
\address{Université Grenoble Alpes, Institut Fourier\\
F-38000 Grenoble, France}
\email{francois.dahmani@univ-grenoble-alpes.fr }
\urladdr{https://www-fourier.ujf-grenoble.fr/~dahmani/}
\author[\initial{M.} \lastname{Mj}]{\firstname{Mahan} \lastname{Mj}}
\address{Tata Institute of Fundamental Research\\
1, Homi Bhabha Road, Mumbai-400005, India}
\email{mahan@math.tifr.res.in}
\email{mahan.mj@gmail.com}
\urladdr{http://www.math.tifr.res.in/~mahan/}
\subjclass{20F65, 20F67, 22E40}
\keywords{Quasiconvex subgroups, hyperbolic groups, relatively
hyperbolic groups, height, convex cocompact subgroups}
\thanks{The first author acknowledge support of ANR grant
ANR-11-BS01-013, and from the Institut Universitaire de France. The
research of the second author is partially supported by a DST J. C. Bose Fellowship. The authors were supported in part by the National Science Foundation under Grant No. DMS-1440140 at the Mathematical Sciences Research Institute in Berkeley during Fall 2016, where this research was completed.}
\begin{abstract}
We introduce the notions of geometric height and graded (geometric) relative hyperbolicity in this paper. We use these to characterize quasiconvexity in hyperbolic groups, relative quasiconvexity in relatively hyperbolic groups, and convex cocompactness in mapping class groups and $\mathrm{Out}(F_n)$.
\end{abstract}
\alttitle{Hauteur, hyperbolicité relative graduée, et quasiconvexité}
\begin{altabstract}
Nous introduisons les notions de hauteur géométrique d'un sous-groupe, et d'hyperbolicité relative graduée d'un groupe, avec une version géométrique de cette dernière. Nous utilisons ensuite ces notions pour caractériser la quasiconvexité des sous-groupes des groupes hyperboliques, la quasiconvexité relative des sous-groupes des groupes relativement hyperboliques, et le fait d'être convexe-cocompact dans un groupe modulaire de surface, ou dans un groupe d'automorphismes extérieurs de groupe libre.
\end{altabstract}
\altkeywords{Sous-groupes quasi-convexes, groupes hyperboliques, groupes relativement hyperboliques, groupes convexes cocompacts}
\maketitle
\tableofcontents
\mainmatter
\section{Introduction}
It is well-known that quasiconvex subgroups of hyperbolic groups have
finite height. In order to distinguish this notion from the notion of
\emph{geometric height} introduced later in this paper, we shall call
the former \emph{algebraic height}: Let $G$ be a finitely generated
group and $H$ a subgroup. We say that a collection of conjugates
$\{ g_iHg_i^{-1}\}$, $i=1, \dots, n$ are \emph{essentially distinct} if
the cosets $\{ g_i H\}$ are distinct.
We
say that $H$ has finite \emph{algebraic height} if there exists
$n \in \mathbb{N}$ such that the intersection of any $(n+1)$
essentially distinct conjugates of $H$ is finite. The minimal $n$ for
which this happens is called the \emph{algebraic height} of $H$.
Thus $H$ has algebraic height one if and only if it is almost malnormal.
This
admits a natural (and obvious) generalization to a finite collection
of subgroups $H_i$ instead of one $H$. Thus, if $G$ is a hyperbolic
group and $H$ a quasiconvex subgroup (or more generally if
$H_1, \dots, H_n$ are quasiconvex), then $H$ (or more generally the
collection $\{ H_1, \dots, H_n\}$) has finite algebraic height
\cite{GMRS}. (See \cite{D03, hruska-wise} for generalizations to the
context of relatively hyperbolic groups.) Swarup asked if the
converse is true:
\begin{qn}[\cite{bestvinahp}]
Let $G$ be a hyperbolic group and $H$ a finitely generated
subgroup. If $H$ has finite height, is $H$
quasiconvex? \label{swarup}
\end{qn}
An example of an infinitely generated (and hence non-quasiconvex)
malnormal subgroup of a finitely generated free group was obtained in
\cite{dm} showing that the hypothesis that $H$ is finitely generated
cannot be relaxed. On the other hand, Bowditch shows in
\cite{bowditch-relhyp} (see also \cite[Prop.\,2.10]{mahan-relrig}) the following positive result:
\begin{theorem}[\cite{bowditch-relhyp}]
Let $G$ be a hyperbolic group
and $H$ a subgroup. Then $G$ is strongly relatively hyperbolic with
respect to $H$ if and only if $H$ is an almost malnormal quasiconvex
subgroup. \label{rhqcchar}
\end{theorem}
One of the motivational points for this paper is to extend Theorem
\ref{rhqcchar} to give a characterization of quasiconvex subgroups of
hyperbolic groups in terms of a notion of {\it graded relative
hyperbolicity} defined as follows:
\begin{defn} \label{grh}
Let $G$ be a finitely generated group, $d$
the word metric with respect to a finite generating set and $H$ a
subgroup. Let $\HH_i$ be the collection of intersections of $i$
essentially distinct conjugates of $H$, let $(\HH_i)_0$ be a choice of
conjugacy representatives, and let $\CH_i$ be the set of cosets of
elements of $(\HH_i)_0$.
Let $d_i$ be the metric on $(G,d)$ obtained by
electrifying\footnote{The second author acknowledges the moderating
influence of the first author on the more extremist terminology
{\it electrocution} \cite{mahan-split, mahan-ibdd}} the elements
of $\CH_i$. We say that $G$ is \emph{graded relatively hyperbolic}
with respect to $H$ (or equivalently that the pair $(G,\{H\})$ has
graded relative hyperbolicity) if
\begin{enumerate}
\item $H$ has algebraic height $n$ for some $n \in \natls$,
\item each element $K$ of $\HH_{i-1}$ has a finite relative
generating set $S_K$, relative to
$H\cap \HH_i (:= \{ H\cap H_i : H_i \in \HH_i\})$; further, the
cardinality of the generating set $S_K$ is bounded by a number depending only on $i$ (and not on $K$),
\item $(G,d_i)$ is strongly hyperbolic relative to $\HH_{i-1}$, where each element $K$ of $\HH_{i-1}$ is equipped with the word
metric coming from $S_K$.
\end{enumerate}
\end{defn}
The following is the main theorem of this paper (see Theorem
\ref{hypcharzn} for a more precise statement using the notion of {\it
graded geometric relative hyperbolicity} defined later)
providing a
partial positive answer to Question \ref{swarup} and a generalization
of Theorem \ref{rhqcchar}:
\begin{theorem} Let $(G,d)$ be one of the following:
\begin{enumerate}
\item $G$ a hyperbolic group and $d$ the word metric with respect to
a finite generating set $S$.
\item $G$ is finitely generated and hyperbolic relative to $\PP$,
$S$ a finite relative generating set, and $d$ the word metric with
respect to $S \cup \PP$.
\item $G$ is the mapping class group $\Mod(S)$ and $d$
a metric
that is equivariantly quasi-isometric to the curve
complex $\rCC(S)$.
\item $G$ is $\Out(F_n)$ and $d$
a metric
that is equivariantly quasi-isometric to the free factor
complex $\FF_n$.
\end{enumerate}
Then (respectively)
\begin{enumerate}
\item if $H$ is quasiconvex, then $(G,\{H\})$ has graded
relative hyperbolicity; conversely, if $(G,\{H\})$ has
geometric graded
relative hyperbolicity then $H$ is
quasiconvex,
\item if $H$ is relatively quasiconvex then $(G,\{H\},d)$ has
graded relative hyperbolicity; conversely, if $(G,\{H\},d)$ has
geometric graded relative hyperbolicity then $H$ is relatively quasiconvex,
\item if $H$ is convex cocompact in $\Mod(S)$ then
$(G,\{H\},d)$ has graded relative hyperbolicity;
conversely, if $(G,\{H\},d)$ has geometric graded relative
hyperbolicity and the action of
$H$ on the curve complex is uniformly proper,
then $H$ is convex cocompact in $\Mod(S)$,
\item if $H$ is convex cocompact in $\Out(F_n)$ then
$(G,\{H\},d)$ has graded relative hyperbolicity;
conversely, if $(G,\{H\},d)$ has geometric graded relative
hyperbolicity and the action of
$H$ on the free factor complex is uniformly proper, then
$H$ is convex cocompact in $\Out(F_n)$.
\end{enumerate}
\end{theorem}
\subsection*{Structure of the paper}
In Section 2,
we will review the notions of hyperbolicity for metric
spaces relative to subsets. This will be related to the notion of hyperbolic embeddedness
\cite{DGO}. We will need to generalize the notion of hyperbolic embeddedness in \cite{DGO} to one of
coarse hyperbolic embeddedness in order to accomplish this. We will also prove
results on the preservation of quasiconvexity under
electrification. We give two sets of proofs: the first set of
proofs relies on assembling diverse pieces of literature on relative hyperbolicity, with several minor
adaptations. We also give a more self-contained set of
proofs relying on asymptotic cones.
In Section 3.1 and the preliminary discussion in Section 4, we give an account of two notions
of height: algebraic and geometric. The classical (algebraic) notion of height of a subgroup
concerns the number of conjugates that can have infinite
intersection. The notion of geometric height is similar, but instead of
considering infinite intersection, we consider unbounded intersections
in a (not necessarily proper) word metric. This naturally leads us to dealing with intersections
in different contexts:
\begin{enumerate}
\item Intersections of conjugates of subgroups in a proper $(\Gamma, d)$ (the Cayley graph of the ambient
group with respect to a finite generating set).
\item Intersections of metric thickenings of cosets in a not necessarily
proper $(\Gamma, d)$.
\end{enumerate}
The first is purely group theoretic (algebraic) and the last
geometric.
Accordingly, we have two notions of height: algebraic and
geometric.
In line with this, we investigate two notions of graded relative
hyperbolicity in Section 4 (\cf Definition \ref{ggrh}):
\begin{enumerate}
\item Graded relative hyperbolicity (algebraic).
\item Graded geometric relative hyperbolicity.
\end{enumerate}
In the fourth section, we also introduce and study a qi-intersection
property, a property that ensures that quasi-convexity is preserved under passage
to electrified spaces. The property
exists in both variants above.
In the fifth and the sixth sections, we will prove our main results
relating height and geometric graded relative hyperbolicity. On a first reading, the
reader is welcome to keep the simplest (algebraic or group-theoretic) notion in mind.
To get a hang of where the paper is headed, we suggest that the reader take a first look at
Sections 5 and 6, armed with Section 3.3 and the
statements of Proposition \ref{prop;hyp_qc_have_fgh},
Theorem \ref{relht},
Theorem \ref{alght-mcg0},
Theorem \ref{alght-out} and
Proposition \ref{prop;satisfiesqiip}. This, we hope, will clarify our intent.
\subsubsection*{Acknowledgments} This work was initiated during a visit of the second author to Institut Fourier in Grenoble during June 2015 and carried on while visiting Indian Statistical
Institute, Kolkata. He thanks the Institutes for their hospitality.
We thank the referee for a detailed and careful reading of the manuscript and for several extremely helpful and perceptive comments.
\section{Relative hyperbolicity, coarse hyperbolic
embeddings}\label{rh}
We shall clarify here what it means in this paper for a
geodesic space $(X,d)$, to be hyperbolic relative to a family
of subspaces $\YY=\{Y_i, i\in I\}$, or to cast it in another
language, what it means for the family $\YY$ to be
hyperbolically embedded in $(X,d)$. There are slight
differences from the more usual context of groups and
subgroups (as~in \cite{DGO}), but we will keep the descending
compatibility (when these notions hold in the context of
groups, they hold in the context of spaces).
We begin by recalling relevant constructions.
\subsection{Electrification by cones} \label{sec;elec}
Given a metric space $(Y, d_Y)$, we will endow $Y\times\nobreak [0,1]$
with the following product metric: it is the largest metric
that agrees with~$d_Y$ on $Y\times \{0\}$,
and each $\{y\} \times [0,1]$ is endowed with a metric isometric to
the segment $[0,1]$.
\begin{defn}[\cite{farb-relhyp}]
Let $(X,d)$ be a geodesic
length space, and $$\YY=\{Y_i,\,i\in I\}$$ be a collection of
subsets of $X$. The \emph{ electrification }
$(X^\rmel_\YY, d^\rmel_{\YY})$ of $(X,d)$ along $\YY$ is
defined as the following coned-off space:
$$\textstyle X^\rmel_\YY = X\sqcup \bigl\{\bigsqcup_{i\in I} Y_i \times [0,1]\bigr\} /\sim$$
where $\sim$ denotes the identification of
$ Y_i\times \{0\}$ with $Y_i\subset X$ for each $i$, and the
identification of $Y_i\times \{1\}$ to a single cone point
$v_i$ (dependent on $i$).
The metric $d^\rmel_{\YY}$ is defined as the path metric on
$X^\rmel_\YY$ for the natural quotient metric coming from the
product metric on $Y_i \times [0,1]$ (defined as above).
\end{defn}
Let $Y_i\in \YY$. The \emph{angular metric} $\hat{d}_{Y_i}$ (or
simply, $\hat{d}$, when there is no scope for confusion) on
$Y_i$ is defined as follows:
For $y_1,y_2 \in Y_i$,
$\hat{d}_{Y_i} (y_1,y_2)$ is the infimum of lengths of paths
in $X^\rmel_\YY$ joining $y_1$ to~$y_2$ not passing through the
vertex $v_i$. (We allow the angular metric to take on infinity as a value).
If $(X,d)$ is a metric space, and $Y$ is a subspace, we write
$d|_Y$ the metric induced on $Y$.
\begin{defn}\label{he}
Consider a geodesic metric space $(X,d)$ and a family of subsets
$\YY =\{Y_i, i\in I\}$. We will say that $\calY$ is \emph{coarsely
hyperbolically embedded} in $(X,d)$, if there is a function
$\psi: \bbR_+\to \bbR_+$ which is proper (\ie
$\lim_{+\infty} \psi(x) = +\infty$), and such that
\begin{enumerate}
\item the electrified space $X^\rmel_\YY$ is hyperbolic,
\item the angular metric at each $Y \in \YY$ in the cone-off is
bounded from below by $\psi \circ d|_Y$.
\end{enumerate}
\end{defn}
\begin{rmark}
This notion originates from Osin's \cite{OsinIJAC}, and was
developed further in \cite{DGO} in the context of groups, where one
requires that each subset $Y_i \in \calY$ is a proper metric space
for the angular metric. This automatically implies the weaker
condition of the above definition. The converse is not true: if
$\YY$ is a collection of uniformly bounded subgroups of a group $X$
with a (not necessarily proper) word metric, it is always coarsely
hyperbolically embedded, but it is hyperbolically embedded in the
sense of \cite{DGO} only if it is finite.
\end{rmark}
As in the point of view of \cite{OsinIJAC}, we say that $(X,d)$ is \emph{strongly hyperbolic relative to} the collection $\calY$ (in the
sense of spaces) if $\calY$ is coarsely hyperbolically embedded in
$(X,d)$.
As we described in the remark, unfortunately, it happens that some
groups (with a Cayley graph metric) are hyperbolic relative to some
subgroups in the sense of spaces, but not in the sense of groups.
Note that there are other definitions of relative hyperbolicity for
spaces. Dru\c{t}u introduced the following definition: a metric space
is hyperbolic relative to a collection of subspaces if all asymptotic
cones are tree graded with pieces being ultratranslates of asymptotic
cones of the subsets.
\Subsection{Quasiretractions}
\begin{defn}
If $(X,d)$ is a metric space, and $Y\subset X$ is a subset endowed
with a metric $d_Y$, we say that $d_Y$ is $\lambda$-undistorted in
$(X,d)$ if for all $y_1, y_2 \in Y$,
$$\lambda^{-1} d(y_1,y_2) -\lambda \leq d_Y(y_1, y_2) \leq \lambda
d(y_1,y_2) +\lambda. $$
We say that $d_Y$ is undistorted in $(X,d)$ if it is
$\lambda$-undistorted in $(X,d)$ for some $\lambda$.
\end{defn}
For the next proposition, define the $D$-coarse path metric on a
subset $Y$ of a path-metric space $(X,d)$ to be the metric on $Y$
obtained by taking the infimum of lengths over paths for which any
subsegment of length $D$ meets $Y$.
The next proposition is the translation (to the present context) of
Theorem 4.31 in \cite{DGO}, with a similar proof.
\begin{prop}\label{retraction}
Let $(X,d)$ be a graph. Assume that $\YY$ is coarsely hyperbolically
embedded in $(X,d)$. Then, there exists $D_0$ such that for all
$Y\in \YY$, the $D_0$-coarse path metric on $Y\in \YY$ is
undistorted (or equivalently the $D_0$-coarse path metric on
$Y\in \YY$ is quasi-isometric to the metric induced from $(X,d)$).
\end{prop}
We will need the following lemma, which originates in Lemma 4.29 in
\cite{DGO}. The proof is the same; for convenience we will briefly
recall it. The lemma provides quasiretractions onto hyperbolically
embedded subsets in a hyperbolic space.
\begin{lemma} Let $(X, d)$ be a geodesic metric space. There exists
$C>0$ such that whenever $\YY$ is coarsely hyperbolically embedded
(in the sense of spaces) in $(X,d)$, then for each $Y\in \YY$, there
exists a map $r: X\to Y$ which is the identity on $Y$ and such that
$\hat{d} (r(x), r(y)) \leq C d(x,y)$.
\end{lemma}
\begin{proof}
Let $p$ the cone point associated to $Y$, and for each $x$, choose a
geodesic $[p,x]$ and define $r(x)$ to be the point of $[p,x]$ at
distance $1$ from $p$. Then $r(x)$ is in~$Y$, and to prove the
lemma, one only needs to check that there is $C$ such that if
$d(x,y)=1$, then $\hat{d} (r(x), r(y)) \leq C $. The constant $C$
will be $10(\delta+1) +1$.
Assume that $x$ and $y$ are at distance $>5(\delta+1)$ from the cone
point. By hyperbolicity, one can find two points in the triangle
$(p, x, y)$ at distance $2(\delta+1)$ from $r(x), r(y)$ at distance
$\leq 2\delta$ from each other. This provides a path of length at
most $6\delta + 4$. Hence $\hat{d} (r(x), r(y)) \leq 6\delta + 4$.
If~$x$ and~$y$ are at distance $\leq 5(\delta+1)$ from the cone
point, then
\[
\hat{d} (r(x), r(y)) \leq d(r(x),x) + d(x,y) + d(y,r(y))\leq
2\times 5(\delta+1) +1.\qedhere
\]
\end{proof}
We can now prove the proposition.
\begin{proof}
Choose $D_0 = \psi(C) $: for all $y_0, y_1 \in Y$ at distance
$\leq D_0$, their angular distance is at most $ C$ (where $C$ is as given by the
lemma above). Consider any path in $X$ from $y_0$ to $y_1$, call the
consecutive vertices $z_0, \dots, z_n$, and project that path by~$r$. One gets $r(z_0), \dots, r(z_n)$ in $Y$, two consecutive ones
being at distance at most $D_0$. This proves the claim.
\end{proof}
\begin{cor}\label{cor-retraction}
If $(X,d)$ is hyperbolic, and if $\YY$ is coarsely hyperbolically
embedded, then there is $C$ such that any $Y\in \YY$ is
$C$-quasiconvex in $X$.
\end{cor}
\subsection{Gluing horoballs}
Given a metric space $(Y, d_Y)$ , one can construct several models of
combinatorial horoballs over it. We recall a construction
(similar to that of Groves and Manning \cite{GM} for a graph).
We consider inductively on $k \in \mathbb{N}\setminus \{0\}$ the space
$\HH_k(Y) = Y\times [1, k]$ with the maximal metric $d_k$
that
\begin{itemize}
\item induces an isometry of $\{y\} \times [k-1, k]$ with $[0,1]$ for
all $y\in Y$, and all $k\geq 1$,
\item is at most $d_{k-1}$ on $\HH_{k-1}(Y) \subset \HH_k(Y)$,
\item coincides with $2^{-k}\times d$ on $Y\times \{k\}$.
\end{itemize}
Then $\HH(Y)$ is defined as the inductive limit of the $\HH_k(Y)$'s and is called the horoball
over $Y$.
Let $(X,d)$ be a graph, and $\YY$ be a collection of subgraphs (with
the induced metric on each of them). The \emph{horoballification} of
$(X,d)$ over $\YY$ is defined to be the space
$X^h_\YY = X\sqcup \{\bigsqcup_{i\in I} \HH(Y_i)\} /\sim_i$, where $\sim_i$ denotes the identification of the boundary horosphere
of $ \HH(Y_i) $ with $Y_i\subset X$. The metric $d^h_{\YY}$ is
defined as the path metric on $X^h_\YY$.
One can electrify a horoballification $X^h_\calY$ of a space $(X,d)$:
one gets a space quasi-isometric to the electrification $X^\rmel_\YY$
of $X$. We record this observation in the following.
\begin{prop} \label{double-elec}
Let $X$ be a graph, and $\YY$ be a
family of subgraphs. Let $X^\rmel_\YY$ and~$X^h_\YY$ be the
electrification, and the horoballification as above. Let
$(X^{h})^\rmel_{\HH(\YY)}$ be the electrification of $X^h_\YY$ over
the collection of horoballs $\HH(Y_i)$ over $Y_i$, $i\in I$.
Then there is a natural injective map
$X^\rmel_\YY \hookrightarrow (X^{h})^\rmel_{\HH(\YY)}$ which is the
identity on~$X$ and sends the cone point of $Y_i$ to the cone
point of $\HH(Y_i)$.
Consider the map $e: ((X^{h})^\rmel_{\HH(\YY)})^{(0)} \to X^\rmel_\YY$
that
\begin{itemize}
\item is the identity on $X$,
\item sends each vertex of $\HH(Y_i)$ of depth $> 2$ to
$v_i \in X^\rmel_\YY$,
\item sends each vertex $(y,n) \in \HH(Y_i)$ of depth $n\leq 2$ to
$y \in Y_i \subset X$,
\item sends the cone point of $((X^{h})^\rmel_{\HH(\YY)})$ associated
to $\HH(Y_i)$ to the cone point of~$X^\rmel_\YY$ associated to
$Y_i$.
\end{itemize}
Then $e$ is a quasi-isometry that induces an isometry on
$X^\rmel_\YY \subset (X^{h})^\rmel_{\HH(\YY)}$.
\end{prop}
\begin{proof}
First, note that a geodesic in $(X^{h})^\rmel_{\HH(\YY)}$ between two
points of $X$ never contains an edge with a vertex of depth
$\geq 1$. If it did, the subpath in the corresponding horoball would
be either non-reduced, or would contain at least $3$ edges, and
could be shortened by substituting a pair of edges through the cone
attached to that horoball. Thus the image of such a geodesic under
$e$ is a path of the same length. In other words, there is an
inequality on the metrics
$d_{ X^\rmel_\YY} \leq d_{(X^h)^\rmel_{\HH(\YY)} }$ (restricted to
$X^\rmel_\YY$). On the other hand, there is a natural inclusion
$X^\rmel_\YY \subset (X^{h})^\rmel_{\HH(\YY)} $, and therefore on
$X^\rmel_\YY$, $ d_{(X^h)^\rmel_{\HH(\YY)} } \leq d_{ X^\rmel_\YY}$.
Thus $e$ is an isometry on $X^\rmel_\YY$. Also, every point in
$ (X^{h})^\rmel_{\HH(\YY)} $ is at distance at most $2$ from a point
in $X$, hence also from a point of the image of $X^\rmel_\YY$.
\end{proof}
\subsection{Relative hyperbolicity and hyperbolic embeddedness}
Recall that we say that a subspace $Q$ of a geodesic metric space
$(X,d)$ is $C$-quasiconvex, for some number $C>0$, if for any two
points $x,y \in Q$, and any geodesic $[x,y]$ in $X$, any point of
$[x,y]$ is at distance at most $C$ from a point of $Q$.
\begin{defn}[{\cite[Def.\,3.5]{mahan-ibdd}}]
A collection
$\mathcal{H}$ of (uniformly) $C$-quasiconvex sets in a
$\delta$-hyperbolic metric space $X$ is said to be \emph{mutually
D-cobounded} if for all $H_i, H_j \in \mathcal{H}$, with
$H_i\neq H_j$, $\pi_i (H_j)$ has diameter less than $D$, where
$\pi_i$ denotes a nearest point projection of $X$ onto $H_i$. A
collection is \emph{mutually cobounded } if it is mutually
D-cobounded for some $D$. \label{cobdd}
\end{defn}
The aim of this subsection is to establish criteria for hyperbolicity
of certain spaces (electrification, horoballification), and related
statements on persistence of quasi-convexity in these spaces. We also
show that hyperbolicity of the horoballification implies strong
relative hyperbolicity, or coarse hyperbolic embeddedness.
Two sets of arguments are given. In the first set of arguments, the
pivotal statement is of the following form: Electrification or
de-electrification preserves the property of being a quasigeodesic.
The arguments are essentially existent in some form in the literature,
and we merely sketch the proofs and refer the reader to specific
points in the literature where these may be found.
The second set of arguments uses asymptotic cones (hence the axiom of
choice) and is more self-contained (it relies on
Gromov-Cartan-Hadamard theorem). We decided to give both these
arguments so as to leave it to the the reader to choose according to
her/his taste.
\Subsection{Persistence of hyperbolicity and quasiconvexity}
\subsubsection{The statements}\label{thestatements}
Here we state the results for which we give arguments in the following
two subsubsections.
\begin{prop}\label{prop;criterion}
Let $(X,d)$ be a hyperbolic geodesic space, $C>0$, and $\YY$ be a
family of $C$-quasiconvex subspaces. Then $X^\rmel_\YY$ is
hyperbolic. If moreover the elements of $\YY$ are mutually
cobounded, then $X^h_\YY$ is hyperbolic.
\end{prop}
In the same spirit, we also record the following statement on
persistence of quasi-convexity.
\begin{prop}\label{cobpersists}
Given $\delta , C$ there exists $ C'$ such that if
$(X,d_X)$ is a $\delta$-hyperbolic metric space with a collection
$\mathcal{Y}$ of $C$-quasiconvex,
sets.
then the following holds:
If $Q (\subset X)$ is a $C$-quasiconvex set (not necessarily an
element of $\YY$), then $Q$ is $C'$\nobreakdash-quasiconvex in
$(X^\rmel_\YY , d_e)$.
\end{prop}
Finally, there is a partial converse. We need a little bit of
vocabulary. If $Z$ is a subset of a metric space $(X,d)$, a $(d,R)$-coarse path in
$Z$ is a sequence of points of $Z$ such that two consecutive points are
always at distance at most $R$ for the metric $d$.
Let $H$ and $Y$ two subsets of $X$. We will denote by $H^{+\lambda}$
the set of points at distance at most $\lambda$ from $H$.
We will say
that $H$ $(\Delta,\epsilon)$-meets $Y$ if there are two points
$x_1,x_2$ at distance $\geq \Delta$ from each
other, and at distance $\leq \epsilon \Delta$ from
$Y$ and $H$, and if for all pair of points at distance $20\delta$
from $\{x_1,x_2\}$, either $H$ or $Y$ is at distance
at least $\epsilon \Delta -2\delta$ from one of them.
The two points $x_1, x_2$ are called a pair of meeting points in
$H$ (for $Y$). We shall say that a subset $H$ of $X$ is coarsely path connected
if there exists $c \geq 0$ such that the
$c$-neighborhood $N_c(H)$ is path connected.
\begin{prop}\label{prop;unfolding_qc}
Let $(X,d)$ be hyperbolic, and let $\calY$ be a collection of
uniformly quasiconvex subsets. Let $H$ be a subset of $X$ that is
coarsely path connected, and quasiconvex in the electrification
$X_\calY^\rmel$.
Assume also that there exists $\epsilon\in (0,1)$, and $\Delta_0$
such that for all $\Delta> \Delta_0$, wherever~$H$
$(\Delta,\epsilon)$-meets an item $Y$ in $\calY$, there is a path in
$H^{+\epsilon \Delta}$ between the meeting points in $H$ that is
uniformly a quasigeodesic in the metric $(X,d)$.
Then $H$ is quasiconvex in $(X,d)$. The quasiconvexity constant of $H$ can be chosen to depend only on
the constants involved for $(X,d), \calY, \Delta_0, \epsilon$, the
coarse path connection constant, and the quasi-geodesic constant of
the last assumption.
\end{prop}
\subsubsection{Electroambient quasigeodesics}
We recall here the concept of electroambient quasigeodesics from
\cite{mahan-ibdd,mahan-split}.
Let $(X,d)$ be a metric space, and $\YY$ a collection of subspaces.
If $\gamma$ is a path in $(X,d)$, or even in
$(X^\rmel_\YY, d^\rmel_{\YY})$, one can define an \emph{elementary
electrification} of $\gamma$ in $(X^\rmel_\YY,
d^\rmel_{\YY})$ as follows:
For $x_1, x_2$ in $\gamma$, both belonging to some $Y_i\in \YY$, and
at distance $>1$, replace the arc of $\gamma$ between them by a pair
of edges $(x_1, v_i) (v_i,x_2)$, where $v_i$ is the cone-point
corresponding to $Y_i$.
A \emph{complete electrification} of $\gamma$ is a path obtained after
a sequence of elementary electrifications of subarcs, admitting no
further elementary electrifications.
One can {\it de-electrify} certain paths. Given a path $\gamma$ in
$(X^\rmel_\YY, d^\rmel_{\YY})$, a \emph{de-electrification} of $\gamma$
is a path $\sigma$ in $(X,d)$ such that
\begin{enumerate}
\item $\gamma$ is a complete electrification of $\sigma$,
\item $(\sigma \setminus \gamma) \cap Y_i$ is either empty or a
geodesic in $Y_i$.
\end{enumerate}
A \emph{$(\lambda,\mu)$-de-electrification} of a path $\gamma$ in
$(X^\rmel_\YY, d^\rmel_{\YY})$, is a path in $X$ such that
\begin{enumerate}
\item $\gamma$ is a complete electrification of $\sigma$,
\item $(\sigma \setminus \gamma) \cap Y_i$ is either empty or a
$(\lambda,\mu)$-quasigeodesic in $Y_i$.
\end{enumerate}
Observe that, given a path $\sigma$ in $X^\rmel$, there might be
several ways to de-electrify it, but these ways differ only in the
choice of the geodesic (or the quasi-geodesic) in the family of
subspaces $Y_i$ corresponding to the successive cone points $v_{Y_i}$
on the path $\sigma$. It might also happen that there is no way of
de-electrifying it, if the spaces in $\YY$ are not quasiconvex.
We say that a path $\gamma$ in $(X,d)$ is an \emph{electroambient
geodesic} if it is a de-electrification of a geodesic.
We say that it is a \emph{$(\lambda, \mu)$-electroambient
quasigeodesic } if it is the $(\lambda, \mu)$-de-electrification of
a $(\lambda, \mu)$-quasigeodesic in $(X^\rmel_\YY, d^\rmel_{\YY})$.
We begin by discussing Proposition \ref{prop;criterion}.
\begin{proof}
The first part is fairly well-known. In some other guise it appears
in \cite[Prop.\,7.4]{bowditch-relhyp} \cite[Prop.\,1]{Szcz} \cite{mahan-ibdd}. In the first two, the electrification by
cones is replaced by collapses of subspaces (identifications to
points) which of course requires that the subspaces to electrify are
disjoint and separated.
However this is only a technical assumption (as explicated in
\cite{mahan-ibdd}). Indeed, by replacing (or augmenting) any
$Y\in \bbY$ by $Y\times [0,D]$ glued along $Y\times \{0\}$, and
replacing~$\YY$ by the family $\{ Y\times \{D\}, Y\in \YY\}$, we
achieve a $D$-separated quasiconvex family.
\end{proof}
Next, we discuss Proposition \ref{cobpersists}.
\begin{proof}
The proofs of Lemma 4.5 and Proposition 4.6 of \cite{farb-relhyp}, Proposition 4.3 and
Theorem 5.3 of \cite{klarreich} (see also \cite{bowditch-relhyp})
furnish Proposition \ref{cobpersists}.
The crucial ingredient in all these proofs is the fact
that in a hyperbolic space, nearest point projections decrease distance exponentially.
Farb proves this in the setup of horoballs in complete simply connected manifolds of pinched
negative curvature. Klarreich ``coarsifies'' this assertion by generalizing it to the context of
hyperbolic metric spaces.
\end{proof}
The rest of this (subsub)section is devoted to discussing Proposition
\ref{prop;unfolding_qc}. Towards doing this, we will obtain an
argument for showing a variant of the second point of Proposition
\ref{prop;criterion}, namely that in a hyperbolic space $(X,d)$, a
family $\YY$ of uniformly quasi convex subspaces that is mutually
cobounded defines a strong relative hyperbolic structure on
$(X,d)$. The second point of \ref{prop;criterion} as it is stated will
be however proved in the next subsection.
We shall have need for the following Lemma \cite[Lem.\,3.9]{mahan-ibdd}
(see also \cite[Prop.\,4.3]{klarreich} \cite[Lem.\,2.5]{mahan-split}).
\begin{lemma}
Suppose $(X,d)$ is $\delta$-hyperbolic. Let $\mathcal{H}$ be a
collection of $C$\nobreakdash-quasiconvex $D$-mutually cobounded subsets. Then for all $P \geq 1$,
there
exists $\epsilon_0 = \epsilon_0 (C, P, D, \delta )$ such that the
following holds:
Let $\beta$ be an electric $(P,P)$-quasigeodesic without backtracking
(\ie $\beta$ does not return to any $H_1 \in \HH$ after leaving it)
and $\gamma$ a geodesic in $(X,d)$, both joining $x, y$.
Then, given
$\epsilon \geq \epsilon_0$
there exists $D = D(P, \epsilon )$ such that
\begin{enumerate}
\item \emph{Similar Intersection Patterns 1:} if precisely one of
$\{ \beta , \gamma\}$ meets an $\epsilon$\nobreakdash-neigh\-borhood
$N_\epsilon (H_1)$ of an electrified quasiconvex set
$H_1 \in \mathcal{H}$, then the length (measured in the intrinsic
path-metric on $N_\epsilon (H_1)$ ) from the entry point to the
exit point is at most $D$,
\item \emph{Similar Intersection Patterns 2:} if both
$\{ \beta , \gamma\}$ meet some $N_\epsilon (H_1)$ then the
length (measured in the intrinsic path-metric on
$N_\epsilon (H_1)$ ) from the entry point of $\beta$ to that of
$\gamma$ is at most $D$; similarly for exit points.
\end{enumerate}
\label{farb2A}
\end{lemma}
Note that Lemma \ref{farb2A} above is quite general and does not
require $X$ to be proper. The two properties occurring in Lemma \ref{farb2A}
were introduced by Farb \cite{farb-relhyp} in the context of a group $G$ and a collection $\HH$
of cosets of a subgroup $H$. The two together are termed `bounded coset penetration'
in \cite{farb-relhyp}.
\begin{remark} In \cite{mahan-ibdd}, the extra hypothesis of
separatedness was used. However, this is superfluous by the same
remark on augmentations of elements of $\YY$ that we made in the
beginning of the proof of Proposition \ref{prop;criterion}. Lemma
\ref{farb2A} may be stated equivalently as the following
(compare with \ref{prop;from_horo_to_he} below) .
If $X$ is a hyperbolic metric space and $\mathcal{H}$ a
collection of uniformly quasiconvex mutually cobounded subsets, then
$X$ is strongly hyperbolic relative to the collection~$\mathcal{H}$.
\end{remark}
We give a slightly modified version of \cite[Lem.\,3.15]{mahan-ibdd}
below by using the equivalent hypothesis of strong relative
hyperbolicity (\ie Lemma \ref{farb2A}).
\begin{lemma} Let $(X,d)$ be a $\delta$-hyperbolic metric space, and
$\HH$ a family of subsets such that $X$ is strongly hyperbolic
relative to $\HH$. Then for all $\lambda, \mu >0$, there exist
$\lambda',\mu'$ such that any electroambient
$(\lambda,\mu)$-quasi-geodesic is a $(\lambda',\mu')$-quasi-geodesic
in $(X,d)$. \label{ea-strong}
\end{lemma}
The proof of Lemma \ref{ea-strong} goes through mutatis mutandis for
strongly relatively hyperbolic spaces as well, \ie hyperbolicity of
$X$ may be replaced by relative hyperbolicity in Lemma \ref{ea-strong}
above. We state this explicitly below:
\begin{cor}\label{ea-cor}
Let $(X,d)$ be strongly relatively hyperbolic relative to a
collection~$\YY$ of path connected subsets. Then, for all
$\lambda, \mu >0$, there exists $\lambda',\mu'$ such that any
electroambient $(\lambda,\mu)$-quasi-geodesic is a
$(\lambda',\mu')$-quasi-geodesic in $(X,d)$.
\end{cor}
We include a brief sketch of the proof-idea following
\cite{mahan-ibdd}. Let $\gamma$ be an electroambient quasigeodesic.
By lemma, its electrification $\hat{\gamma}$ is a quasi-geodesic
in~$X^\rmel_\YY$. Let~$\sigma$ be the electric geodesic joining the
end-points of $\gamma$. Hence $\sigma$ and $\hat{\gamma}$ have
similar intersection patters with the sets $Y_i$ \cite{farb-relhyp},
\ie they enter and leave any $Y_i$ at nearby points. It then suffices
to show that an electroambient representative of $\sigma$ is in fact
a quasigeodesic in $X$. A proof of this is last statement is given in
\cite[Th.\,8.1]{ctm-locconn} in the context of horoballs in
hyperbolic space (see also Lemmas 4.8, 4.9 and their proofs in
\cite{farb-relhyp}). The same proof works after horoballification for
an arbitrary relatively hyperbolic space.\qed
The proof of Proposition \ref{prop;unfolding_qc} as stated will be given in the next subsubsection.
We shall provide here a proof that suffices for the purposes of this paper. We assume, in addition
to the hypothesis of the proposition that there exists an integer $n>0$, and $ D_0 \geq 0$
such that for all distinct $Y_1, \dots, Y_n
\in \YY$, $\bigcap_i Y_i^{+\epsilon}$ has diameter at most~$D$. The existence of such a
number $n$ will translate into
the notion of finite geometric height later in the paper.\vspace*{-8pt}\enlargethispage{.5\baselineskip}%
\begin{proof} We prove the statement by inducting on $n$. For $n=1$ there is nothing to
show; so we start with $n=2$. Note that in this case, the hypothesis is equivalent
to the assumption that the $Y_i$'s are cobounded.
Assume therefore that the elements of $\YY$ are uniformly quasiconvex in
$(X,d)$; and that they are uniformly mutually cobounded.
We shall show that $H$ is also quasiconvex for a uniform
constant.
First, since $(X,d)$ is
hyperbolic
it follows by Proposition \ref{prop;criterion}
that $(X^\rmel_\YY,d^\rmel)$ is hyperbolic.
Let $x, y \in H$. By
assumption, there exists $C_0 \geq 0$ such that $H$ is
$(C_0,C_0)$-qi embedded in $(G, d^\rmel)$. Denote by $\PP$ the set of
cone points corresponding to elements of ${\YY}$ and let $\gamma$ be
a $(C,C)$-quasi-geodesic without backtracking in $(X,d^\rmel)$ with
vertices in $H \cup \PP$ joining $x,y \in H$. By assumption, the
collection ${\YY}$ is uniformly $C$\nobreakdash-quasiconvex. Further, by
assumption, there exists $\epsilon\in (0,1)$, and $\Delta_0$ such
that for all $\Delta> \Delta_0$, wherever $H$
$(\Delta,\epsilon)$-meets an item $Y$ in $\calY$, there is a path in
$H^{+\epsilon \Delta}$ between the meeting points in $H$ that is
uniformly a quasigeodesic in the metric $(X,d)$. Hence, for some
uniform constants $\lambda, \mu$, we may (coarsely)
$(\lambda,\mu)$-de-electrify $\gamma$ to obtain a
$(\lambda,\mu)$-electroambient quasigeodesic $\gamma'$ in $(X,d)$,
that lies close to $H$.
[Note that the meeting points of $H$ with elements of $\YY $ are
only coarsely defined. So we are actually replacing pieces of
$\gamma$ by quasigeodesics in $H^{+\epsilon \Delta}$ rather than in
$H$ itself.]
Note that by assumption $\YY$ are uniformly quasiconvex in
$(X,d)$; and further that they are uniformly mutually cobounded. Hence
the space $X$ is actually strongly hyperbolic
relative to $\YY$.
By Corollary \ref{ea-cor}, it follows that $\gamma'$ is a
quasi-geodesic in $(X, d)$, for a uniform constant.
Since this was done for arbitrary $x,y \in H_{i,\ell}$, we obtain
that $H $ is $D$-quasiconvex in $(X,d)$. This finishes the proof of
Proposition \ref{prop;unfolding_qc} for $n=2$.
The induction step is now easy. Assume that the statement is true for $n=m$. We shall prove
it for $n=m+1$. Electrify all pairwise intersections of $Y_i^{+\epsilon}$ to obtain an electric metric
$d_2$. Then the collection $\{ Y_i\}$ is cobounded with
respect to the electric metric $d_2$.
Here again, the space
$(X,d_2)$ is strongly hyperbolic relative to
the collection $\{ Y_i\}$. By
the argument in the case $n=2$ above, $H$ is quasiconvex in $(X,d_2)$. The collection of
pairwise intersections of the $Y_i^{+\epsilon}$'s in $X$ satisfies the property that an intersection
of any $m$ of them is bounded. We are then through by the induction hypothesis.
\end{proof}
\subsubsection{Proofs through asymptotic cones}
The repeated use of different references coming from different
contexts in the previous subsubsection might call for more systematic
self-contained proofs of the statements of subsection
\ref{thestatements}. This is our purpose in this subsubsection.
In this part we will use the structure of an argument originally due to
Gromov, and developed by Coulon amongst others (see for instance
\cite[Prop.\,5.28]{Coulon_IJAC}), which uses asymptotic cones in
order to show hyperbolicity or quasiconvexity of constructions.
We fix a non-principal ultrafilter $\omega$
and will use the
construction of asymptotic cones with respect to this ultrafilter $\omega$. A few
observations are in order here.
In all the following, $(X_N,x_N)$ is a sequence of pointed
$\delta_N$-hyperbolic spaces, with~$\delta_N$ converging to
$0$. Recall then that
the asymptotic cone $\lim_\omega (X_N,x_N)$ is an $\mathbb{R}$-tree,
with a base point.
If $\YY_N$ is a family of $c_N$-quasiconvex subsets of
$X_N$ (for $c_N$ tending to $0$), we want to
consider the asymptotic cone $\lim_\omega
((X_N)_{\YY_N}^\rmel, x_N)$ and relate it to $\lim_\omega
(X_N,x_N)$.
Let us define the following equivalence relation on the set of
sequences in $X_N$. Two sequences $(u_N), (v_N)$ are
equivalent if $d_{X_N} (u_N,v_N) = O(1)$ for the
ultrafilter $\omega$ (more precisely, if there exists a
constant $C$ such that for $\omega$-almost all values of
$N$, $d_{X_N} (u_N,v_N) \leq C$). Let us consider
the set of equivalence classes of sequences, and let us only
keep those that have some (hence all) representative $(u_N)$
such that the electric distance $d_{X_N^\rmel}(x_N, u_N) $ is
$O(1)$. Let us call $\frakC$ this set of equivalence
classes. For a sequence $u=(u_N)$ we write $\sim_u$ for its class
in $\frakC$. We also allow ourselves to write $\lim_\omega (X_N,
\sim_u)$ for $\lim_\omega (X_N, u_N)$ to avoid cluttered notation.
\begin{lemma} \label{lem;manycopies}
There is a natural inclusion from the disjoint union
$$\bigsqcup_{\sim_u \in \frakC} \lim_\omega (X_N, \sim_u)$$
into $\lim_\omega ((X_N)_{\YY_N}^\rmel, x_N)$.
\end{lemma}
\begin{proof}
By definition of $\frakC$ we have a well defined map
$$\lim_\omega (X_N, \sim_u) \to \lim_\omega
((X_N)_{\YY_N}^\rmel, x_N)$$ for each class $\sim_u$ in $\frakC$.
Given two sequences $(y_N)$, and $(z_N)$ in the same class
$\sim_u\in \frakC$, if $d_{X_N}(z_N,y_N)$ is not $o(1)$
for $\omega$, then in the electric metric, it is not
$o(1)$, since the added edges all have length $1$. Thus
this map is injective. If $(y_N)$, and $(z_N)$ are not in
the same class in $\frakC$, then $d_{X_N}(z_N,y_N)$ is not
$o(1)$ for $\omega$, and again, in the electric metric, it is not
$o(1)$. The map of the lemma is thus injective.
\end{proof}
Note that the inclusion is continuous, as inclusions along
the sequence are distance non-increasing. But it is not isometric. We need to
describe what happens with the cone off.
Consider a sequence $(Y_N)$ of subsets of $\calY$. One says
that the sequence is visible in $\lim_\omega (X_N, u_N)$
if $d_{X_N} (u_N, Y_N) \leq O(1)$. In that case,
$\lim_\omega (Y_N, u_N)$ is a subset of $\lim_\omega (X_N,
u_N)$, consisting of the images of all the sequences of elements of $Y_N$
that remain at $O(1)$-distance from $u_N$. Note that given
a sequence $(Y_N)$, it can be visible in several limits
$\lim_\omega (X_N, u_N)$ (for several non-equivalent
$(u_N)$). In those classes where $(Y_N)$ is not visible,
$\lim_\omega (Y_N,
u_N) $ is empty.
Let us define $\lim_\omega (Y_N, *)$
to be $\bigsqcup_{\sim_u \in \frakC} \lim_\omega (Y_N, u_N)$
in $\bigsqcup_{\sim_u \in \frakC} \lim_\omega (X_N, u_N)$. By
the previous lemma, this is naturally a subset of
$\lim_\omega((X_N)^{\it el}_{\calY_N}, x_N)$.
We define $\calY^\omega$ to be the collection of all sets
$\lim_\omega (Y_N, *)$, for all possible sequences $(Y_N)
\in \prod_{N>0} Y_N$. This is a family of subsets of
$\lim_\omega((X_N)^{\it el}_{\calY_N}, x_N)$.
Let us define $\left[\bigsqcup_{\sim_u \in \frakC} \lim_\omega (X_N,
u_N)\right]^\rmel$ to be the cone-off (of parameter $1$) of
$\left[\bigsqcup_{\sim_u \in \frakC} \lim_\omega (X_N,
\sim_u)\right]$ over each $\lim_\omega (Y_N, *)$. Note
that there is a natural copy of $\left[\lim_\omega (X_N,
\sim_u)\right]^\rmel$ in $\left[\bigsqcup_{\sim_u \in \frakC} \lim_\omega (X_N,
\sim_u)\right]^\rmel$.
\begin{lemma} \label{lem;electrification_of_ascone}
$\lim_\omega ((X_N)_{\YY_N}^\rmel, x_N)$ is isometric to $\left[\bigsqcup_{\sim_u \in \frakC} \lim_\omega (X_N, \sim_u)\right]^\rmel$.
\end{lemma}
\begin{proof}
First there is a natural bijection
\[
\lim_\omega((X_N)_{\YY_N}^\rmel, x_N)\isom\biggl[\bigsqcup_{\sim_u \in \frakC}\lim_\omega (X_N, \sim_u)\biggr]^\rmel.
\]
Indeed, for any
sequence $(y_N)$ with $y_N\in X_N$ at distance $O(1)$ from $x_N$ in
the electric metric, Lemma \ref{lem;manycopies} provides an image in
$\left[\bigsqcup_{\sim_u \in \frakC} \lim_\omega (X_N,
\sim_u)\right]^\rmel$. For any sequence~$(c_N)$ with $c_N$ a cone-point in
$X_N^{\it el}\setminus X_N$, at distance $O(1)$ from $x_N$ in
the electric metric, $c_N$ is in the cone electrifying a certain
$Y_N$, which is therefore at distance $O(1)$ from $x_N$ for the
electric metric. Choose $u_N\in Y_N$, then the equivalence class of
$(u_N)$ is in $\frakC$, and of course $Y_N$ is visible in this
class. Thus, there is a cone point $c \in \left[\bigsqcup_{\sim_u \in
\frakC} \lim_\omega (X_N, \sim_u)\right]^\rmel$ at distance
$1$ from $\lim_\omega(Y_N, *)$. We choose this point as the image of
the sequence $(c_N)$ in $\lim_\omega ((X_N)_{\YY_N}^\rmel,
x_N)$. This is well defined, and injective, for if $c'_N$ is another
sequence of cone-points $\omega$-almost everywhere different from
$c_N$, then it defines an $\omega$-almost everywhere different
sequence $Y'_N$, and a different set $\lim_\omega(Y'_N, *)$. We also
can extend our map to all $\lim_\omega ((X_N)_{\YY_N}^\rmel, x_N)$
linearly on the cone-edges. This produces a bijection $\lim_\omega
((X_N)_{\YY_N}^\rmel, x_N) \to \left[\bigsqcup_{\sim_u \in \frakC}
\lim_\omega (X_N, \sim_u)\right]^\rmel$.
To show that it is an isometry, consider two sequences $(y_N), (z_N)$
both in $X_N$,
such that the distance in $X_{\YY_N}^\rmel$ converges (for $\omega$) to
$\ell$. Then there is a path of length~$\ell_N$ (converging to $\ell$)
in $X_{\YY_N}^\rmel$ with, eventually, at
most $\ell/2$ cone points on it. It follows from the construction that
their images in $\left[\bigsqcup_{\sim_u \in \frakC}
\lim_\omega (X_N, \sim_u)\right]^\rmel$ are at distance at most
$\ell$.
Conversely, assume $(y_N)$ and $(z_N)$ are sequences
in $X_N$
giving points
in $\lim_\omega (X_N, \sim_u)$ and $\lim_\omega (X_N,
\sim_v)$, for $\sim_u, \sim_v \in\frakC$, and take a path $\gamma$ between
these points in $\lim_\omega
((X_N)_{\YY_N}^\rmel, x_N) \simeq \left[\bigsqcup_{\sim_u \in \frakC}
\lim_\omega (X_N, \sim_u)\right]^\rmel$, of length $\ell >0$. It has finite
length, so it contains
finitely many cone points $c_i$ ($i=1, \dots, r$), coning $\lim_\omega
(Y_N^{(i)}, *)$,
for which
$Y_N^{(i)}$ is visible in both $\sim_{u_i}, \sim_{u_{i+1}}$. This easily
produces a path in $\lim_\omega ((X_N)_{\YY_N}^\rmel, x_N)$ of
length $\ell$, by using the corresponding cone points and the path
between the spaces $Y_N^{(i)}$ given by the restriction of $\gamma$.
We have thus observed that the bijection we started with is $1$-Lipschitz as is its inverse. It is therefore an isometry.
\end{proof}
We finally describe a tree-like structure on $\left[\bigsqcup_\frakC \lim_\omega (X_N,
\sim_{u}) \right]^\rmel$ where the pieces are the subspaces
$\left[\lim_\omega (X_N, \sim_u)\right]^\rmel$ for $\sim_u
\in \frakC$, which are
electrifications of $\lim_\omega (X_N, \sim_u)$ over the
subsets of the form $\lim_\omega (Y_N, \sim_u)$, for all sequences
$(Y_N)\in \prod(\calY_N)$ that are visible in
$\sim_u$.
Let us say that two classes $\sim_u$ and $\sim_v$ are joined
by a sequence $(Y_N)$ if the latter sequence is visible in both of
them.
First we describe a simpler case of this tree-like structure.
\begin{lemma} Assume that $\YY_N$ consists of $c_N$-quasiconvex
subsets of $X_N$ with $c_N$ tending to $0$.
For any pair of different classes $\sim_u, \sim_v$ in
$\frakC$, the subspaces
$\left[\lim_\omega (X_N, \sim_u)\right]^\rmel$ and
$\left[\lim_\omega (X_N, \sim_v)\right]^\rmel$ of
$\left[\bigsqcup_{\sim_w\in \frakC} \lim_\omega (X_N,
\sim_{w}) \right]^\rmel$ have
intersection of diameter at most~$2$.
\end{lemma}
\begin{proof}
Note that by Lemma \ref{lem;manycopies} the intersection consists of cone
points. Thus consider two sequences $(Y_N), (Y'_N)$ both visible in
$\sim_u$ and $ \sim_v$. Consider then $y_N(u), y_N(v) \in Y_N$ such
that $y_N(u)$ is visible in $\sim_u$ (hence not in $\sim_v$) and symmetrically
$y_N(v)$ is visible in $\sim_v$ (hence not in $\sim_u$), and take $y'_N(u),
y'_N(v) \in Y'_N$ similarly. The distances $d_{X_N}(y_N(u), y'_N(u))$ and
$d_{X_N}(y_N(v), y'_N(v))$ both are $O(1)$ whereas $d_{X_N}(y_N(u), y_N(v))$ and
$d_{X_N}(y'_N(u), y'_N(v))$ both go to infinity (for $\omega$). The
space $X_N$ being a $\delta_N$-hyperbolic space (for $\delta_N\to 0$),
the quadrilateral with these four vertices have their sides $[y_N(u),
y_N(v)]$ and $[y'_N(u), y'_N(v)]$ getting $o(1)$-close to each
other, on sequences that are visible for $\sim_u$ and sequences that
are visible for $\sim_v$. But these sides are close to $Y_N$ and
$Y'_N$ respectively. It follows that in $\lim_\omega (X_N,
\sim_u)$ and in $\lim_\omega (X_N, \sim_v)$, the limit of
$Y_N$ and of $Y'_N$ share a point. Thus the cone point of their
electrifications are at distance $2$.
\end{proof}
Note that if there is a bound on the diameter of the projection of
$Y_N$ on $Y'_N$, then there is only one point in the intersection.
\begin{lemma}
If there is a cycle of classes $\sim_{u_1}, \sim_{u_2}, \dots,
\sim_{u_k}, \sim_{u_{k+1}}=\sim_{u_1}$, where $\sim_{u_i}$ is joined
to $\sim_{u_{i+1}}$ by a sequence $(Y^{(i)}_N)$, then there is
$11$, so that it is
$\delta_0$-hyperbolic, and such that $\YY$ is a collection of
$C_0$-quasiconvex subsets. Let us define $X_\YY^{el_{\lambda}}$ to
be
$$\textstyle X^{el_{\lambda}}_\YY = X\sqcup \bigl\{\bigsqcup_{i\in I} Y_i \times[0,\lambda]\bigr\} /\sim,$$
where $\sim$ denotes the identification of $ Y_i\times \{0\}$ with
$Y_i\subset X$ for each $i$, and the identification of
$Y_i\times \{1\}$ to a single cone point $v_i$ (dependent on $i$),
and where $Y_i \times [0,\lambda]$ is endowed with the product
metric as defined in the first paragraph of \ref{sec;elec} except
that $\{y\} \times [0,n]$ is isometric to $[0,\lambda]$. The claim
ensures that $X_\YY^{el_{\lambda}}$ is hyperbolic. However, it is
obviously quasi-isometric to $X_\YY^\rmel$. We have the first point.
For the second part, one can proceed with a similar proof, with
horoballs. The claim is then that for all $\rho$, there exist $\delta_0, C_0$
and $D_0$ such that if $X$ is $\delta_0$-hyperbolic, and if $\YY$ is
a collection of $C_0$-quasiconvex subsets, $D_0$-mutually
cobounded, then any ball of radius $\rho$ of the horoballification
$X^h_\YY$ is $10$-hyperbolic.
The proof of the claim is similar. Consider a sequence of
counterexamples $X_N, \YY_N$, for the parameters
$\delta = C = D = 1/N$ for $N$ going to infinity, with the four
points $x_N, y_N, z_N, t_N$ in $(X_N)^h_\YY$, in a ball of radius
$\rho$, falsifying the hyperbolicity condition.
There are two cases. Either $x_N$ (which is in $(X_N)^h_\YY$)
escapes from $X_N$, \ie its distance from some basepoint in $X_N$
tends to $\infty$ for the ultrafilter $\omega$, or it does not. In
the case that it escapes from $X_N$, then, when it is larger than $\rho$
all four points $x_N, y_N, z_N, t_N$ are in a single horoball, but
such a horoball is $10$-hyperbolic hence a contradiction.
The other case is when there is $x'_N\in X_N$ whose
distance to $x_N$ remains bounded (for the ultrafilter $\omega$).
Note that $\{ x_N, y_N, z_N, t_N\}$ converge in the asymptotic cone
$\lim_\omega ((X_N)_{\YY_N}^h, x'_N )$ of the sequence of pointed
spaces $( (X_N)_{\YY_N}^h, x'_N)$. It is also immediate by
definition of $\lim_\omega \YY_N$ that
$\lim_\omega ((X_N)_{\YY_N}^h, x'_N )$ is the horoballification of
the asymptotic cone of the sequence $(X_N, x'_N)$ over the family
$\lim_\omega (\YY_N,x'_N)$ defined above.
This family $\lim_\omega (\YY_N, x'_N)$ consists of \emph{convex}
subsets (hence subtrees), such that any two share at most one point.
This horoballification is therefore a tree-graded space in the sense
of \cite{DrutuSapir}, with pieces being the combinatorial horoballs
over the subtrees constituting $\lim_\omega \YY_N$. As a tree of
$10$\nobreakdash-hyperbolic spaces, this space is $10$\nobreakdash-hyperbolic, contradicting
the inequalities satisfied by the limits
$\lim_\omega \{ x_N, y_N, z_N, t_N\}$. Therefore, $X_\YY^h$ is
$\rho$-locally $10$-hyperbolic. As before, one may check that (under the same assumptions) $X^h_\YY$
is \hbox{$(2+10C_0+10\delta_0)$}-coarsely simply connected, and again this
implies by the Gromov-Cartan-Hadamard theorem that (under the same
assumptions) $X^h_\YY$ is hyperbolic.
This implies the second point. Indeed, let us denote by
$\frac{1}{\lambda} X$ the space $X$ with metric rescaled by
$ \sfrac{1}{\lambda}$.
The previous claim shows that, under the assumption of the second
point of the proposition, there exists $\lambda>1$ such that
$\lambda (\frac{1}{\lambda} X)^{h}_{\frac{1}{\lambda}\YY} $ is
hyperbolic. Consider the map $\eta$ between
$ X^{h}_\YY \to \lambda (\frac{1}{\lambda}
X)^{h}_{\frac{1}{\lambda}\YY} $
that is identity on $X$ and that sends $\{y\} \times \{n\}$ to
$\{y\}\times \{ \lambda \times ( n + \lfloor \log_2 \lambda \rfloor
)\}$
for all $y\in Y_i$ and all $Y_i$ (and all $n$). All paths in
$X_\YY^h$ that have only vertical segments in horoballs have their
length expanded (under the map $\eta$) by a factor between $1$
and $\lambda+\log_2\lambda$. But the geodesics in $X_\YY^h$ and
$ \lambda (\frac{1}{\lambda} X)^{h}_{\frac{1}{\lambda}\YY} $ are
paths whose components in horoballs consist of a vertical
(descending) segment, followed by a single edge, followed by a
vertical ascending segment (see \cite{GM}).
Hence
$\eta$ is a quasi-isometry, and the space $ X^{h}_\YY $ is
hyperbolic.
\end{proof}
We continue with the persistence of quasi-convexity of Proposition \ref{cobpersists}.
\begin{proof}
The strategy is similar to that in the previous proposition. The
main claim is that for all $\rho$, there is $\delta_0<1, C_0<1$ such
that if $(X,d_X)$ is $\delta_0$-hyperbolic, if $\mathcal{Y}$ is a
collection of $C_0$-quasiconvex subsets and if $Q$ is another
$C_0$-quasiconvex subset of $X$, then $Q$ is $\rho$-locally
$10$-quasiconvex in $X^\rmel_\YY$ (of course $\delta_0, C_0$ will be
very small).
To prove the claim, again, by contradiction, consider a sequence
$X_N, \YY_N, Q_N$ of counter examples for $\delta_N=C_N= 1/N$ for
$N=1,2,\dots$. There exist two points $x_N, y_N$
in $Q_N$, at distance $\leq \rho$ from each other (for
the electric metric), and a geodesic
$ [ x_N, y_N ]$ in $X^\rmel_\YY$ with a point $z_N$ on it at distance
$>10$ from $Q$. We record a point $z'_N$ in $Q$ at minimal distance
($\leq \rho $ in any case) from $z_N$.
With a non principal ultrafilter $\omega$, we may take the
asymptotic cone of the family of pointed spaces $(X_N^\rmel, x_N)$. In
$\lim_\omega ((X_N)^\rmel_\YY,x_N)$, the sequences $(y_N)$,
$([x_N, y_N])$ and $(z_N)$ have limits for which the distance
inequalities persist, and we get that $\lim_\omega (Q_N,x_N)$ is not
$\rho$-locally $10$-quasiconvex in
$\lim_\omega ((X_N)^\rmel_\YY,x_N)$. But as we noticed in
Corollary \ref{coro;quasitree}
$\lim_\omega ((X_N)^\rmel_\YY,x_N)$ is a 2-quasi-tree of
spaces of the form
$( \lim_\omega (X_N, x'_N))^\rmel_{\YY^\omega}$, which are the
electrifications of $\mathbb{R}$-trees $\lim_\omega
(X_N, x'_N)$ over a
family of convex subsets (\ie subtrees).
In this space, $\lim_\omega (Q_N, x_N)$
is also a subforest of
$\bigsqcup_{(u_N)\in \frakC} \lim_\omega (X_N, u_N)$.
Also observe that if $(Q_N)$ is visible in two adjacent
classes, then $\lim_\omega (Q_N, x_N)$ is adjacent to their common cone point
over sequences $ (Y_N^{(i)}), (Y_N^{(j)}) $.
Hence $\lim_\omega (Q_N, x_N)$ is $2$-quasiconvex in
$ \lim_\omega ((X_N)^\rmel_{ \YY^\omega}, x_N)$, and this
contradicts the inequalities satisfied by
$\lim_\omega \{x_N, y_N, z_N, z'_N\}$. The claim is established for
all $\rho$.
Now there exists $\rho_0$ such that, in any $1$-hyperbolic space,
any subset that is $\rho_0$-locally $10$-quasiconvex is
$10^{14}$-globally quasiconvex (this classical fact, perhaps found
elsewhere with other (better!) constants, follows also from the
Gromov-Cartan-Hadamard theorem for instance). So, by choosing an
appropriate $\rho$, we have proven that there is $\delta_0<1, C_0<1$
and $C_1$, such that if $(X,d_X)$ is $\delta_0$-hyperbolic, if
$\mathcal{Y}$ is a collection of $C_0$-quasiconvex subsets and if
$Q$ is another $C_0$-quasiconvex subset of $X$, then $Q$ is
$C_1$-quasiconvex in $X^\rmel_\YY$.
Coming back to the statement of the proposition, by rescaling our
space, we have proven that if $(X,d_X)$ is a $\delta$-hyperbolic
metric space with a collection $\mathcal{Y}$ of
$C$\nobreakdash-quasiconvex
sets, and if $Q$ is $C$-quasiconvex, then $Q$ is
$\lambda C_1$-quasiconvex in $X_\YY^{el_\lambda}$ (as~defined in the
previous proof) for $\lambda = \max\{ \delta/\delta_0, C/C_0\}$.
Since $X_\YY^{el_\lambda}$ is quasi-isometric to $X_\YY^\rmel$, by a
$(\lambda,\lambda)$-quasi-isometry, it follows that $Q$ is
$C'$-quasiconvex in~$X_\YY^\rmel$ for $C'$ depending only on
$\delta, C$.
\end{proof}
Finally, we consider the proposed converse in Proposition \ref{prop;unfolding_qc}. We use the same strategy again. The claim is now the
lemma below.
To state it, we need to define the $m$-coarse path
metric on an $m$-coarse path connected subspace of a metric
space. A subset $Y\subset X$ of a metric space
is $m$-coarse path connected if for any two
points $x,y$ in it there is a sequence $x_0=x, x_1, \dots,
x_r = y$ for some $r$ such that $x_i\in Y$ and $d_X(x_i,
x_{i+1}) \leq m$ for all $i$. We call such a sequence an
$m$-coarse path or a path with mesh $\leq m$. The length of the coarse
path $(x_0, \dots, x_r)$ is $\sum d_X(x_i,
x_{i+1})$. The $m$-coarse path metric on $Y$ is the distance
obtained by taking the infimum of lengths of
coarse paths between its points. An $m$-coarse geodesic is a
coarse path realizing the coarse path metric between two
points.
\begin{lemma}
Fix $C_H^\rmel$, $R>0$, $Q>1$, $\epsilon >0$, and
$\Delta >10\epsilon$.
Then there exists $\delta_0, C_0, m_0 >0$, such that the
following holds:
Assume that $(X,d)$ is geodesic,
$\delta_0$-hyperbolic, with a collection $\calY$ of
$C_0$-quasiconvex subsets. Further suppose that
$H$ is an $m_0$-coarsely connected subset of $X$ which is
$C_H^\rmel$\nobreakdash-quasiconvex in the electrification $X_\calY^\rmel$.
Equip $H^{+\epsilon\Delta}$ with its $m_0$-coarse path metric~$d_H$.
Assume also that whenever $H$ $(\Delta, \epsilon)$-meets a set
$Y \in \calY$, there is a $(Q, C_0)$\nobreakdash-quasi\-geodesic
path,
which is an $(m_0/10)$-coarse path, in $H^{+\epsilon \Delta}$ joining the meeting points
in $H$.
Then for all $a,b \in H^{+\epsilon \Delta}$ at $d_H$-distance at
most $R$ from each other, any $m_0$-coarse $\delta_0$-quasi-geodesic
of $H^{+\epsilon \Delta}$ (for its coarse path metric) between $a,b$ is
$(\Delta\times C_H^\rmel)$-close to a geodesic of $X$.
\end{lemma}
\begin{proof}
Suppose that the claim is false: For all choice of $\delta, C, m$ there is
a counterexample.
Set $\delta_N = C_N = 1/N$.
For each $\epsilon$, there exists $N$ such that, in a
$(\sfrac{1}{N})$-hyperbolic space, for any two points $x,y$, and any
$(Q, 1/N)$-quasigeodesic $p$ which is a $(1/N)$-coarse path between these two
points, the $\epsilon$-neighborhood of $p$ contains
the geodesics $[x,y]$. (This, for instance, is visible on an
asymptotic cone).
Thus, it is possible to choose a sequence $m_N>10/N$ decreasing to
zero, such that pairs of $(\sfrac{9\Delta }{10})$-long
$(Q, 1/N)$-quasigeodesics with mesh $\leq 1/N$ in a
$(\sfrac{1}{N})$\nobreakdash-hyperbolic spaces, with starting points at distance
$\leq \Delta/10$ from each other, and ending points at distance
$\leq \Delta/10$ from each other, necessarily lie at distance
$(m_N/10)$ from one another.
Let then $X_N, H_N, \calY_N$ be a counterexample to our claim for
these values: for each~$N$ there is $a_N, b_N$ in $H_N^{+\Delta}$,
$R$-close to each other for $d_{H_N}$, and a point
$c_N\in H_N^{+\Delta}$ in a coarse $\delta_N$-quasi-geodesic in
$[a_N, b_N]_{d_{H_n}}$ at distance at least
$(\Delta\times C_H^\rmel)$ from a geodesic $[a_N, b_N]$ in $X_N$.
However $c_N$ is $ C_H^\rmel$-close to a geodesic $[a_N, b_N]_\rmel$
in $X^\rmel$. Passing to an asymptotic cone, we find a map
$p_\omega$ from an interval $[0,R]$ to a continuous path
in $\lim_\omega (H_N, a_N)$ from $a^{\omega}$
to $b^{\omega}$
(which can be equal to $a^{\omega}$) that passes through a point~$c^{\omega}$ at distance $\geq (\Delta\times C_H^\rmel)$ from the arc
$[a^{\omega}, b^{\omega}]$ in $\lim_\omega(X_N, a_N)$.
However,
it is at distance $\leq C_H^\rmel$ in the electrification of
$\lim_\omega(X_N, a_N)$ by $\calY^{\omega}$. It follows that on the path in $\lim_\omega(X_N, a_N)$ from $[a^{\omega}, b^{\omega}]$ to
$c^{\omega}$, there must exist a segment
of length $\geq \Delta$
belonging to the same
$Y^\omega \in \calY^{\omega}$ . Let us say
that $Y^\omega$ is the limit of a sequence $Y_N$. Note that the
limit path $p_\omega$ crosses this segment at least twice (once in
either direction).\enlargethispage{.5\baselineskip}%
Thus, for $N$ large enough, $H_N$ $(\Delta,\epsilon)$-meets $Y_N$,
with two pairs of meeting points $(r_1, r_2), (s_1, s_2)$ in $H_N$, where $d(r_1,r_2) \geq 9\Delta/10$ (and $(s_1,s_2) \geq 9\Delta/10$)
and $d(r_1,s_1) \leq 3\Delta/10$ and $ d(r_2,s_2) \leq 3\Delta/10$.
By assumption, there is a $(Q, \sfrac{1}{N})$-quasi-geodesic path in
$H^{+\epsilon \Delta}$ from $s_1$ to $s_2$ and another from $r_1$ to
$r_2$, with mesh $\max\{100 \epsilon, \Delta_0\}$. Let~$Q$ be the
quasi-geodesic constant given by the the assumption of the
proposition on $(\Delta,\epsilon)$\nobreakdash-meetings, and $C_H^\rmel$ be as given
by the assumption. Take $R$ larger (how large will be made clear in the proof).
Rescale the space $X$ so that the hyperbolicity constant, the
quasiconvexity constant of items of $\calY$, and the constant of
coarse path connection of $H$ are respectively smaller than
$\delta_0, C_0, m_0$ of the lemma above.
Note that the assumption of the proposition on
$(\Delta,\epsilon)$-meetings is invariant under rescaling (except for
the value of $\Delta_0$). Thus, this assumption still holds, with the
same~$\epsilon$, and for the specified $\Delta$ chosen above. The lemma applies, and $H^{+\epsilon \Delta}$ is $R$\nobreakdash-locally
quasiconvex for the rescaled metric. By the local to global principle
(in $\delta_0$ hyperbolic spaces), with a suitable preliminary (large enough) choice
of $R$, $H^{+\epsilon \Delta}$ is then globally quasiconvex. After rescaling back to the
original metric of $X$, $H^{+\lambda}$ is still quasiconvex for some
$\lambda$ (depending on $\epsilon\Delta$, and the coefficient of
rescaling); hence~$H$ is quasiconvex.
By construction, we also have the statement on the dependence of the
quasiconvexity constant.\qed
\subsubsection{Coarse hyperbolic embeddedness and strong relative
hyperbolicity}
The following proposition establishes the equivalence of coarse
hyperbolic embeddedness and strong relative hyperbolicity in the
context of this paper.
\begin{prop}\label{prop;from_horo_to_he}
Assume that $(X,d)$ is a metric space, and that $\YY$ is a
collection of subspaces.
If the horoballification $X_\YY^h$ of $X$ over $\YY$ is hyperbolic,
then $\YY$ is coarsely hyperbolically embedded in the sense of
spaces.
If $X$ is hyperbolic and if $\YY$ is coarsely hyperbolically
embedded in the sense of spaces, then $X_\YY^h$ is hyperbolic.
\end{prop}
We remark here parenthetically that the converse should be true
without the assumption of hyperbolicity of $X$. However, this is not
necessary for this paper.
\begin{proof}
Assume that the horoballification $X_\YY^h$ of $X$ over $\YY$ is
$\delta$-hyperbolic. The horoballs $Y^h$ (corresponding to $Y$) are
thus $10\delta$-quasiconvex. Therefore, by Proposition
\ref{prop;criterion}, the electrified space obtained by electrifying
(coning off) the horoballs $Y^h$ is hyperbolic.
Since by Proposition \ref{double-elec} this space is
quasi-isometric to $(X^\rmel_\YY, d^\rmel_\YY)$, it follows that the
later is hyperbolic. This proves the first condition of Definition
\ref{he}.
We want to prove the existence of a proper increasing function
$\psi:\bbR_+\to\bbR_+$, such that the angular metric at each cone
point $v_Y$ (for $Y\in \YY$) of $(X^\rmel_\YY, d^\rmel_\YY)$ is
bounded below by $\psi \circ d|_Y$. Define
$$ \psi(r) = \inf_{Y\in \YY} \inf_{\substack{y_1, y_2 \in Y\\ d(y_1, y_2)\leq r}} \hat{d}(y_1, y_2). $$
Of course, the angular metric at $v_Y$ is bounded below by
$\psi \circ d|_Y$. The function $\psi$ is obviously increasing. We
need to show that it is proper, \ie that it goes to $+\infty$.
If $\psi$ is not proper, then there exists $\theta_0 > 0 $ such that
for all $D$, there exist $Y\in \YY $ and $y, y' \in Y$ at
$d$-distance greater than $D$ but $\hat{d} (y,y') \leq \theta_0$
(where $\hat{d}$ is the angular metric on $Y$). We choose
$D \gg\theta_0\delta$ (for instance $D= \exp(100 (\theta_0+1)(\delta+1))$).
Consider a path in $(X^h_\YY)^\rmel$ of length less than $\theta_0$
from $y$ to $y'$ avoiding the cone point of $Y$. Because
$D\gg\theta_0$, this path has to pass through other cone points. It
can thus be chosen as a concatenation of $N+1$ geodesics whose
vertices are $y,y'$ and some cone points $v_1, \dots, v_N$
(corresponding to $Y_1, \dots, Y_N$ with $N<\theta_0$). Adjoining
the (geodesic) path $[y,v_Y] \cup [v_Y, y']$ (where $v_Y$
corresponds to the cone point for $Y$), we thus have a geodesic
$(N+2)$-gon $\sigma$.
Next replace each passage of $\sigma$ through a cone point ($v_i$ or
$v_Y$) in $X^\rmel_\YY$ by a geodesic ($\mu_i$ or $\mu_Y$
respectively) in the corresponding horoball ($Y_i^h$ or $Y^h$
respectively) in $X_\YY^h$ to obtain a geodesic $(2N+2)$-gon $P$ in
$X_\YY^h$. The geodesic segments $\mu_i$ or $\mu_Y$ comprise $(n+1)$
alternate sides of this geodesic $(2N+2)$-gon.
Since $X_\YY^h$ is $\delta$-hyperbolic, it follows that the
mid-point $m$ of $\mu_Y$ is at distance
$\leq (2N+2)\delta$ from another edge of
$P$. Note that $m$ is in the horoball of $Y$, and because the
distance in $Y$ between $y$ and $y'$ is larger than $D$, we have
that $d^h(m, Y) $ is at least $\log(D)/2$.
Since no other edge of $P$ enters the horoball $Y^h$, this forces
$\log(D)$ (and hence $D$) to be bounded in terms of $\theta_0$
and $\delta$: $D\leq \exp(4 (N +1) \delta)$. Since $N\leq \theta_0$,
this is a
contradiction with the choice of $D$. We can conclude that $\psi$ is
proper, and we have the first statement.
Let us consider the second statement. If $X$ is hyperbolic and if
$\YY$ is coarsely hyperbolically embedded in the sense of spaces,
then elements of $\YY$ are uniformly quasiconvex in $(X,d)$ by
\ref{cor-retraction},
and, by the property of the angular distance on any $Y\in \YY$, they
are mutually cobounded.
The statement then follows by Proposition \ref{prop;criterion}.
\end{proof}
\section{Algebraic height and intersection properties}
\subsection{Algebraic height}
We
recall here the general definition for height of finitely many subgroups.
\begin{defn} Let $G$ be a group and $\{ H_1, \dots, H_m\}$ be a finite collection of subgroups.
Then the \emph{algebraic height} of this collection is
$n$ if
$(n+1)$ is the smallest number with the property that
for any $(n+1)$ distinct left cosets $g_1H_{\alpha_1},\dots,
g_{n+1}H_{\alpha_{n+1}}$, the intersection $\bigcap_{1\leq i\leq n+1} g_i
H_{\alpha_i} g_i^{-1}$ is finite.
\end{defn}
We shall describe this briefly by saying that algebraic height is the largest $n$ for which the intersection of
$n$ \emph{essentially distinct} conjugates of $ H_1, \dots, H_m$ is infinite. Here `essentially distinct' refers to
the cosets of $ H_1, \dots, H_m$ and not to the conjugates themselves.
For hyperbolic groups, one of the main theorems of
\cite{GMRS} is the following:
\begin{theorem}[\cite{GMRS}]\label{alght}
Let $G$ be a hyperbolic group and $H$ a quasiconvex subgroup. Then the algebraic height of $H$ is finite.
Further, there exists $R_0$ such that if $H \cap gHg^{-1} $ is infinite, then $g$ has a double coset representative with length at most~$R_0$.
The same conclusions hold for finitely many quasiconvex subgroups $\{
H_1, \dots, H_n\}$ of $G$.
\end{theorem}
We quickly recall a proof of Theorem \ref{alght} for one subgroup $H$
in order to generalize it to the context of mapping class groups and $\Out(F_n)$.
\begin{proof}
Let $G$ be hyperbolic, $X(=\Gamma_G)$ a Cayley graph of
$G$ with respect to a finite generating set
(assumed to be $\delta$-hyperbolic), and $H$ a
$C_0$-quasiconvex subgroup of~$G$.
Suppose that there exist $N$ essentially distinct
conjugates $\{ H^{g_i}\}$, $i=1, \dots, N$, of $H$ that
intersect
in an infinite subgroup. The $N$ left-cosets
$g_iH$ are disjoint and
share an accumulation point $p$ in the boundary of $G$ (in the limit set
of $\bigcap_i H^{g_i}$). Since all~$g_iH$ are $C_0$-quasiconvex, there exist
$N$ disjoint quasi-geodesics $\sigma_1, \dots, \sigma_N$
(with same constants $\lambda, \mu$
depending only on $C_0, \delta$) converging to $p$ such
that $\sigma_i$ is in~$g_iH$.
Since~$X$ is $\delta$-hyperbolic, there exists $R
(=R(\lambda, \mu, \delta) = R(C_0, \delta))$ and a point~$p_0$
sufficiently far along $\sigma_1$ such that all the
quasi-geodesics $\sigma_1, \dots, \sigma_N$ pass through
$B_R(p_0)$. Hence $N \leq \#(B_R(p_0))$
giving us finiteness of height.
Further, any such $\sigma_i$ furnishes a double coset representative
$g_i^\prime$ of $g_i$ (say by taking a word that gives the shortest
distance between
$H$ and the coset $g_iH$) of length bounded in terms of
$R$. This furnishes the second conclusion.
\end{proof}
\begin{remark}
A word about generalizing the above
argument to a family $\HH$ of finitely many subgroups
is necessary.
The place in the above argument where $\HH$ consists of
a singleton is used essentially is in declaring that
the
$N$ left-cosets
$g_iH$ are disjoint. This might not be true in general
(e.g. $H_1 < H_2$ for a family having two elements).
However, by the pigeon-hole principle, choosing $N_1$ large enough, any
$N_1$ distinct conjugates $\{ H_i^{g_i}\}$, $i=1, \dots,
N$, $H_i \in \HH$ must contain $N$
essentially distinct conjugates $\{ H^{g_i}\}$, $i=1,
\dots, N$ of some $H \in \HH$ and then the
above argument for a single $H\in \HH$ goes through.
\end{remark}
\begin{remark}
A number of other examples of finite
algebraic height may be obtained
from certain special subgroups of
relatively hyperbolic groups, mapping class groups and
$\Out(F_n$). These will be discussed after we introduce
geometric height later in the paper.
\end{remark}
\subsection{Geometric $i$-fold intersections}
Given a finite family of subgroups of a group we define
collections of geometric $i$-fold intersections.
\begin{defn}\label{def:ifoldinter}
Let $G$ be endowed with a left invariant word metric
$d$. Let $\HH$ be a finite family of subgroups of $G$.
For $i\in \bbN, i\geq 2$, define
the \emph{geometric $i$-fold intersection}, or simply the $i$-fold intersection of cosets of $\HH$, $\HH_i$, to be the set of subsets
$J$ of $G$ for which there exist $H_1, \dots, H_i \in
\HH$ and $g_1, \dots, g_i \in G$, and $\Delta \in \mathbb{N}$ satisfying:
$$ J= \biggl( \bigcap_j \left( g_jH_j \right)^{+\Delta} \biggr) $$
and $ \bigcap_j \left( g_jH_j \right)^{+\Delta} $ is not in the $20\delta$-neighborhood of
$ \bigcap \left( g_jH_j \right)^{+\Delta-2\delta}
$, and the diameter of $J$ is at least
$10\Delta$.
\end{defn}
Geometric $i$-fold intersections are thus, by definition,
intersections of thickenings of cosets. The condition that the
diameter of the intersection is larger than $10$ times the thickening
is merely to avoid counting myriads of too small intersections.
The next proposition establishes that the collection of such intersections is again closed under intersection.
\begin{prop}\label{geointnsstable}
Consider $J\in \HH_j$, and $K\in \HH_k$ for $j\max (\Delta_J, \Delta_K)$.
Assume that $J$ and $K$
$(\Delta,\epsilon)$-meet,
for some $\Delta>20\Delta_0$, and for $\epsilon <1/50$.
Then either $K\subset J$, or for any pair of
$(\Delta, \epsilon)$-meeting points of $J$ and $K$,
there is $L\in \HH_{j+1}$ contained in $J$,
that contains it.
\end{prop}
\begin{proof}
Let $x,y $ be $(\Delta, \epsilon)$-meeting points of $J$ and $K$. If $K\not\subset J$,
we can assume that $x,y$ are in $\left( g'_1H'_1 \right)^{+\Delta_K+\epsilon\Delta} $ for some $g'_1H'_1 $ not contained in the collection $ \{ g_iH_i\}$.
Notice that $x,y$ are in $ \bigcap_i \left( g_iH_i \right)^{+ \Delta_J +\epsilon\Delta} \cap \left( g'_1H'_1 \right)^{+\Delta_K +\epsilon\Delta} $,
hence in $ \bigcap_i \left( g_iH_i \right)^{+\Delta' } \cap \left( g'_1H'_1 \right)^{+ \Delta'} $ for $\Delta'$ the greater of $\Delta_K +\epsilon \Delta$ and $\Delta_J + \epsilon \Delta$.
We argue
by contradiction. Suppose that $x,y$ are contained in the $20\delta$-thickening of a $2\delta$-lesser intersection. It follows that there are $x',y'$ such that $d(x,x')\leq 20\delta$, $d(y,y') <20\delta$ and still $d(x', J) \leq \epsilon \Delta -2\delta$, and $d(y', J) \leq \epsilon \Delta -2\delta$. But by definition of $(\Delta,\epsilon)$-meeting, this is a
contradiction.
Finally,
the diameter of the intersection of $\Delta'$-thickenings of
our cosets, is larger than $ \Delta $.
Since the thickening constant is
$\Delta' \leq \Delta_0+ \epsilon\Delta$, the ratio of
the thickening constant by the diameter is at most $(\Delta_0+
\epsilon\Delta)/\Delta$ which is less than $ 1/10$, hence the
result.
\end{proof}
Let $(G,d)$ be a group with a word metric and $H C+20\delta$, the intersection
$\bigcap A_i ^{+\Delta}$ is $(4\delta)$-quasiconvex in
$(G,d)$.
Moreover, if $A$ and $B$ are
$C$-quasiconvex subsets of $G$, and if $\Pi_B(A)$
denotes the set of nearest points projections
of $A$ on $B$, then, either $\Pi_B(A) \subset A^{+3C
+10\delta}$ or $\diam \Pi_B(A)
\leq 4C+20\delta$.
\end{prop}
\begin{proof}
Consider $x,y \in \bigcap A_i^{+\Delta}$ and take $a_i, b_i$
some nearest point projection on $A_i$. On a
geodesic $[x, y]$ take $p$ at distance greater than $4\delta$
from $x$ and $y$. Hyperbolicity applied to the quadrilateral
$(x, a_i, b_i, y)$ tells us that $x$ is $4\delta$-close to $[x'_i,
a_i]\cup [a_i, b_i] \cup [b_i, y'_i]$, where $x'_i$ and $y'_i$ are the
points of, respectively $[x, a_i]$ and $[y, b_i]$, at distance
$4\delta$ from, respectively, $x$ and $y$.
Let us call $[x'_i,
a_i], [b_i, y'_i] $
the approaching segment, and $[a_i, b_i]$ the traveling
segments. Hence for each $i$, $p$ is closed to either an approaching
segment, or the traveling segment, with subscript $i$.
\begin{itemize}
\item
If $p$ is close to
an approaching segment of index $i$, then it is in $A_i^{+\Delta}$.
\item
If $x$ is close to the traveling segment of index $i$, then it is at
distance at most $C+10\delta$ from $A_i$, hence in $A_i^{+\Delta}$
because $\Delta>C+10\delta$.
\end{itemize}
We thus obtain that $[x,y]$ remains at distance $4\delta$
from $\bigcap A_i^{+\Delta}$.
To prove the second statement, take $a_0, b_0$ in
$A$ and $B$ respectively realizing the distance (up to $\delta$ if
necessary). Let $b \in \Pi_B(A)$, and assume that it is the projection of
$a$. In the quadrilateral $a,a_0,b,b_0$, the geodesic $[a,a_0]$ stays
in $A^{+C}$ and $[b,b_0]$ is in $B^{+C}$. Since $b$ is a
projection, $[b,a]$ fellow-travels $[b,b_0]$ for less than $2C+10\delta$, and
similarly for $[b_0,a_0]$ with $[b_0,b]$. By hyperbolicity $[b,b_0]$
thus stays $10\delta$ close to $[a,a_0]$ except for the part
$(2C+10\delta)$-close to either $b$ or $b_0$.
It follows that either $b \in A^{+(3C +10 \delta)}$ or
$b$ is at distance $\leq 4C+20\delta$ from $b_0$.
Thus $\Pi_B(A) \subset A^{+3C
+10\delta}$
or $\diam \Pi_B(A)
\leq 4C+20\delta$.
\end{proof}
\subsection{Algebraic $i$-fold intersections}
We provide now a more algebraic (group theoretic) treatment
of the preceding discussion.
This is in keeping with the more well-known setup of intersections of subgroups
and their conjugates, \cf \cite{GMRS}.
Given a finite family of subgroups of a group we first
define collections of $i$-fold conjugates
or algebraic $i$-fold intersections.
\begin{defn}\label{essdist}
Let $G$ be endowed with a left invariant word metric
$d$. Let $\HH$ be a family of subgroup of $G$.
For $i\in \bbN, i\geq 2$, define $\HH_i$ to be the set of subgroups
$J$ of~$G$ for which there exists $H_1, \dots, H_i \in
\HH$ and $g_1, \dots, g_i \in G$ satisfying:
\begin{itemize}
\item the cosets $g_j H_j$ are pairwise distinct (and
hence as in \cite{GMRS} we use the terminology that the
conjugates
$\{g_j H_jg_j^{-1}, j=1,\dots,i\}$
are \emph{essentially distinct})
\item $J$ is the intersection $\bigcap_j g_j H_j g_j^{-1}$.
\item $J$ is unbounded in $(G,d)$.
\end{itemize}
We shall call $\HH_i$ the family of \emph{algebraic $i$-fold intersections}
or simply, \emph{$i$-fold conjugates}.
\end{defn}
The second point in the following definition is motivated
by the behavior of nearest point projections of
cosets of quasiconvex subgroups of hyperbolic groups on
each other. Let $(G,d)$ be hyperbolic and $H_1, H_2$
be quasiconvex. Let $aH_1, bH_2$ be cosets and
$c=a^{-1}b$. Then the nearest point projection of $bH_2$
onto $aH_1$ is the (left) $a$\nobreakdash-translate of the nearest
point projection of $cH_2$ onto $H_1$.
Let $\Pi_B (A)$ denote the (nearest-point) projection of
$A$ onto $B$. Then $\Pi_{H_1} (cH_2)$ lies in a bounded
neighborhood (say $D_0$-neighborhood) of $H_2^c\cap H_1$
and so $\Pi_{aH_1} (bH_2)$ lies in a
$D_0$-neighborhood of $bH_2 b^{-1}a \cap\nobreak aH_1$. The latter does lie
in a bounded neighborhood of $(H_2)^b \cap (H_1)^a$, but
this bound depends on $a, b$ and is not uniform. Hence the
somewhat convoluted way of stating the second property
below. The language of nearest-point projections below is
in
the spirit of \cite{mahan-ibdd, mahan-split} while the
notion of geometric $i$-fold intersections discussed earlier
is in the spirit of \cite{DGO}.\enlargethispage{.5\baselineskip}%
\begin{defn} Let $G$ be a group and $d$ a word
metric on $G$.
A finite family $\calH = \{ H_1, \dots , H_m\}$ of
subgroups of $G$, each equipped with a word-metric $d_i$
is said to
have the \emph{uniform qi-intersection property}
if there exist $C_1, \dots, C_n , \dots$ such that
\begin{enumerate}
\item for all $n$, and all $H \in \HH_n$,
$H$ has a conjugate $H'$ such that if $d'$ is any \emph{induced metric} on $H'$ from some $H_i \in \HH$, then
$(H',d')$ is $(C_1,C_1)$-qi-embedded in $(G,d)$,
\item for all $n$, let $(\HH_n)_0$ be a choice of conjugacy representatives
of elements of~$\HH_n$ that are $C_1$-quasiconvex in $(G,d)$;
Let $\CC\HH_n$ denote the collection of left cosets of
elements of $(\HH_n)_0$; for all $A, B \in \CC\HH_{n}$ with $A=aA_0, B=bB_0$,
and $A_0, B_0 \in (\HH_n)_0$,
$\Pi_B (A)$ either has diameter bounded by $C_n$ for the metric
$d$, or
$\Pi_B (A)$ lies in a (left) $a$-translate translate of a
$C_n$-neighborhood of $A_0\cap B_0^c$, where $c=a^{-1}b$.
\end{enumerate}
\label{qiintn}
\end{defn}
In keeping with the spirit of the previous subsection, we
provide a geometric version of the above definition below.
\begin{defn} Let $G$ be a group and $d$ a word
metric on $G$.
A finite family $\calH = \{ H_1, \dots , H_m\}$ of
subgroups of $G$, each equipped with a word-metric $d_i$
is said to
have the \emph{uniform geometric qi-intersection property}
if there exist $C_1, \dots, C_n , \dots$ such that
\begin{enumerate}
\item for all $n$, and all $H \in \HH_n$,
$(H,d)$ is $C_n$-coarsely path connected, and
$(C_1,C_1)$-qi-embedded in $(G,d)$ (for its coarse path metric),
\item for all $A, B \in \HH_{n}$
either $\diam_{G,d} ( \Pi_B (A) ) \leq C_n$, or
$\Pi_B (A) \subset A^{+C_n} $ for $d$.
\end{enumerate}
\label{geoqiintn}
\end{defn}
\begin{remark}{\rm
The second condition of Definition \ref{geoqiintn}
follows from the first condition
if $d$ is hyperbolic by Proposition \ref{prop;projections}.
Further, the first condition holds for such $(G,d)$ so long as $\Delta$ is taken
of the order of the quasiconvexity constants (again by
Proposition \ref{prop;projections}).
Note further that if $G$ is hyperbolic (with respect to
a not necessarily locally finite word
metric) and $H$ is $C$-quasiconvex, then
by Proposition \ref{geointnsstable}
the collection of geometric $n$-fold intersections
$\HH_n$ is mutually cobounded for the metric of
$(G,d)^\rmel_{\HH_{n+1}}$ (as in Definition
\ref{cobdd}). }
\label{rem:cobdd}
\end{remark}
\subsection{Existing results on algebraic intersection properties}
We start with the following result due to Short.
\begin{theorem}[{\cite[Prop.\,3]{short}}] Let $G$ be a
group generated by the finite set $S$.
Suppose $G$ acts properly on a uniformly proper geodesic
metric space $(X,d)$, with a base point $x_0$.
Given $C_0$, there exists $C_1$ such that
if $H_1, H_2$ are subgroups of $G$ for
which the orbits $H_ix_0$ are $C_0$-quasiconvex in $ (X,d)$ (for
$i=1,2$) then the orbit $ (H_1 \cap H_2)x_0$ is $C_1$-quasiconvex in $(X,d)$.
\label{short-intn}
\end{theorem}
We remark here that in the original statement of
\cite[Prop.\,3]{short}, $X$ is itself the Cayley graph
of $G$ with respect to $S$, but the proof there goes
through
without change to the general context of Proposition \ref{short-intn}.
In particular, for $G$ (Gromov) hyperbolic, or
$G=\Mod(S)$ acting on Teichmüller space $\Teich(S)$ (equipped with the Teichmüller metric)
and
$\Out(F_n)$ acting on Outer space $cv_N$ (with the symmetrized Lipschitz metric),
the notions of (respectively) quasiconvex subgroups or
convex cocompact subgroups of $\Mod(S)$ or $\Out(F_n)$ (see Sections \ref{mcg} and
\ref{outfn} below for the lemmas) are
independent of the finite generating sets chosen. Hence
we have the following.
\begin{theorem}
Let $G$ be either $\Mod(S)$ or $\Out(F_n)$
equipped with some finite generating set.
Given $C_0$, there exists $C_1$ such that
if $H_1, H_2$ are $C_0$-convex cocompact subgroups of $G$,
then $H_1 \cap H_2$ is $C_1$-convex cocompact in
$G$. \label{short-intn-coco}
\end{theorem}
The corresponding statement for relatively hyperbolic
groups and relatively quasiconvex groups is due to Hruska.
For completeness we recall it.
\begin{defn}[\cite{osin-relhyp, Hru}]\label{relqc}
Let $G$
be finitely generated hyperbolic relative to a finite
collection $\PP$ of parabolic subgroups.
A subgroup $H \le G$ is \emph{relatively quasiconvex} if the following holds.
Let $S$ be some (any) finite relative generating set for $(G,\PP)$,
and let $P$ be the union of all $P_i \in \PP$.
Let $\overline\Gamma$ denote the Cayley graph of $G$ with
respect to the generating set $S\cup P$
and $d$ the word metric on $G$.
Then there is a constant $C_0=C_0({S},d)$ such that
for each geodesic $\gamma \subset \overline{\Gamma}$
joining two points of $H$,
every vertex of $\gamma$ lies within $C_0$ of~$H$
(measured with respect to $d$).
\end{defn}
\begin{theorem}[\cite{Hru}]
Let $G$ be finitely generated
hyperbolic relative to $\PP$.
Given~$C_0,$ there exists $C_1$ such that if $H_1, H_2$
are $C_0$-relatively quasiconvex subgroups of $G$,
then $H_1 \cap H_2$ is $C_1$-relatively quasiconvex in
$G$. \label{hruska-intn}
\end{theorem}
\section{Geometric height and graded geometric relative hyperbolicity}
We are now in a position to define the geometric analog of
height. There are two closely related notions
possible, one corresponding to the geometric notion of
$i$-fold
intersections and one corresponding to the algebraic notion of
$i$-fold conjugates. The former is relevant when one
deals with subsets and the latter when one
deals with subgroups.
\begin{defn} \label{gh}
Let $G$ be a group, with a left invariant word metric $d
(=d_G)$ with respect to some (not necessarily finite) generating set.
Let $\HH$ be a family of subgroups of $G$.
The \emph{geometric height},
of $\HH$ in $(G,d)$ (for $d$) is the minimal number
$i\geq 0$ so that the collection $\HH_{i+1}$ of $(i+1)$-fold
intersections consists of uniformly bounded sets.
If $H$ is a single subgroup, its geometric height is that of the
family $\{H\}$.
\end{defn}
\skpt
\begin{remark}[Comparing notions of height]\label{comp1}
\begin{itemize}
\item Geometric
height is related to algebraic height, but is more flexible, since
in the former, we allow the group $G$ to have an infinite generating
set.
We are then free to apply the operations of electrification, horoballification
in the context of non-proper graphs.
\item In the case of a locally finite
word metric, algebraic height is less than or equal to geometric
height. Equality holds if all bounded intersections are
uniformly bounded.
\item For a locally finite
word metric, finiteness of algebraic height
implies that $i$-fold conjugates are finite (and hence
bounded
in any metric) for all sufficiently large $i$. Hence
finiteness of
geometric height
follows from finiteness of algebraic height and of a
uniform bound on the diameter of the finite
intersections.
\item When the metric on a Cayley graph is not locally finite,
we do not know of any general statement that allows us to go directly from finiteness
of diameter of an intersection of thickenings of cosets (geometric condition)
to finiteness of diameter of intersections
of conjugates (algebraic condition). Some of the technical complications below are due to this difficulty in
going from geometric intersections to algebraic intersections.
\end{itemize}
\end{remark}
We generalize Definition \ref{grh} of graded relative hyperbolicity
to the context of geometric height as follows.
\begin{defn} \label{ggrh}
Let $G$ be a group, $d$ the word metric with respect to
some (not necessarily finite) generating set
and $\calH$ a finite collection of subgroups.
Let $\HH_i$ be the collection of all
$i$-fold conjugates of $\HH$. Let $(\HH_i)_0$ be a choice of conjugacy
representatives, and $\CC\HH_i$ the set of left cosets of elements
of $(\HH_i)_0$
Let $d_i$ be the metric on $(G,d)$ obtained by electrifying
the elements of $\CC\HH_i$.
Let $\CC\calH_\bbN$ be the graded family
$(\CC\calH_i)_{i\in \mathbb{N}}$.
We say that $G$ is
\emph{graded geometric relatively hyperbolic} with
respect to $\CC\calH_\bbN$
if
\begin{enumerate}
\item $\calH$ has
geometric height $n$ for some $n \in \natls$, and for each $i$
there are finitely many orbits of $i$-fold intersections,
\item for all $i\leq n+1$, $\CC\HH_{i-1}$ is
coarsely hyperbolically embedded in
$(G,d_i)$,
\item there is $D_i$
such that all items of $\CC\HH_i$ are $D_i$-coarsely
path connected in $(G,d)$.
\end{enumerate}
\end{defn}
\begin{remark}[Comparing geometric and algebraic graded relative hyperbolicity] \label{comp2}
Note that
the second condition of
Definition \ref{ggrh} is equivalent, by Proposition
\ref{prop;from_horo_to_he}, to saying that
$(G,d_i)$ is strongly hyperbolic relative to the collection
$\HH_{i-1}$. This is exactly the
third (more algebraic) condition in Definition \ref{grh}.
Also,
the third condition
of Definition \ref{ggrh} is the analog of (and follows from)
the second (more algebraic) condition in Definition \ref{grh}.
Thus finite
geometric height along with (algebraic)
graded relative hyperbolicity implies
graded geometric
relative hyperbolicity.
\end{remark}
The rest of this section furnishes examples of finite
height in both its geometric and algebraic incarnations.
\Subsection{Hyperbolic groups}
\begin{prop}\label{prop;hyp_qc_have_fgh}
Let $(G,d)$ be a hyperbolic group with a locally finite word metric, and
let $H$ be a quasiconvex subgroup of $G$. Then $H$ has finite
geometric height.
More precisely, if $C$ is the quasi-convexity constant of $H$ in
$(G,d)$, and if $\delta$ be
the hyperbolicity constant in $(G,d)$, and if $N$ is the cardinality of
a ball of $(G,d)$ of radius $2C+10\delta$, and if $g_0H, \dots, g_k
H$ are distinct cosets of $H$ for which there exists $\Delta$ such
that the total intersection $\bigcap_{i=0}^k (g_i H)^{+\Delta}$ has diameter more
than $10\Delta$, and more that $100\delta$, then there exists $x \in
G$ such
that each $g_i H$ intersects the ball of radius $N$ around
$x$.
\end{prop}
First note that the second statement implies the first in the
(by the third
point of Remark \ref{comp1}). We will directly
prove the second.
The proof is similar to the finiteness of the algebraic height. Also
note that the second statement can be rephrased in terms of double
cosets representatives of the $g_i$: under the assumption on the total
intersection, and if $g_0=1$, there are double coset
representatives of the $g_i$ of
length at most $2(2C+10\delta)$.
\begin{proof}
Assume that there exists $\Delta>0$, and elements $1=g_0, g_1, \dots, g_k$
for which the cosets $g_i H$ are
distinct, and $\bigcap_{i=0}^k (g_i H)^{+\Delta}$ has diameter larger
than $10\Delta$ and than $100\delta$.
First we treat the case $\Delta >5\delta$. Pick $y_1, y_2 \in \bigcap_{i=0}^k (g_i H)^{+\Delta}$ at distance
$10\Delta$ from each
other, and pick $x\in [y_1, y_2]$ at distance larger
than $\Delta +10\delta$ from both $y_i$. For each $i$ an application
of hyperbolicity and quasi-convexity tells us that $x$ is at distance at
most $2C +10\delta$ from each of $g_i H$. The ball of radius $2C
+10\delta$ around $x$ thus meets each coset $g_iH$.
If $\Delta \leq 5\delta$, we pick $y_1, y_2 \in
\bigcap_{i=0}^k (g_i H)^{+\Delta}$ at distance
$100\delta$ from each
other, and take $x$ at distance greater than $10\delta$ from both
ends. The end of the proof is the same.
\end{proof}
\subsection{Relatively hyperbolic groups}
If $G$ is hyperbolic relative to a collection of subgroups
$\calP$, then Hruska and Wise defined in \cite{hruska-wise}
the relative height of
a subgroup $H$ of $G$ as $n$ if $(n+1)$ is the smallest number with
the property that for any $(n+1)$ elements $g_0, \dots, g_n$ such that
the $g_iH$ are $(n+1)$-distinct cosets, the intersections of
conjugates $\bigcap_i g_i H g_i^{-1}$ is finite or parabolic.
The notion of relative algebraic height is actually the geometric
height for the relative distance, which is given by a word
metric over a generating set that is the
union of a finite set and a set of conjugacy representatives of the
elements of $\calP$.
Indeed, in a relatively hyperbolic group, the
subgroups that are bounded in the relative metric are precisely those
that are finite or
parabolic. We give a quick argument.
It follows from the lemma of relative quasiconvexity
that a subgroup having finite diameter in the electric metric on $G$
(rel. $\PP$) is relatively quasiconvex. It is also true \cite{DGO} that the normalizer of any $P \in \PP$
is itself and that the subgroup generated by any $P$ and any infinite order
element $g\in G \setminus P$ contains the free product of conjugates of $P$ by $g^{kn}, k \in \Z$.
Since any proper supergroup of $P$ necessarily contains such a $g$, it follows that no proper supergroup
of $P$ can be of finite diameter in the electric metric on $G$
(rel. $\PP$). It follows that bounded subgroups are precisely the finite subgroups or
those contained inside parabolic subgroups.
The notion of relative height can
actually
be extended to define the height of a collection of
subgroups $H_1, \dots, H_k$, as in the case for the algebraic height.
Hruska and Wise proved that relatively quasiconvex subgroups have finite relative
height. More precisely:
\begin{theorem}[{\cite[Th.\,1.4, Cors.\,8.5-8.7]{hruska-wise}}]\label{relht}
Let $(G,\PP)$
be relatively hyperbolic, let $S$ be a finite relative
generating set for $G$ and $\Gamma$ be the Cayley graph
of $G$ with respect to $S$.
Then for $\sigma \geq 0$, there exists $C\geq 0$
such that the following holds.
Let $H_1, \dots , H_n$
be a finite collection of $\sigma$-relatively quasiconvex
subgroups of $(G,\PP)$.
Suppose that there exist distinct cosets $\{
g_mH_{\alpha_m}\}$ with $\alpha_m \in \{1, \dots ,
n\}$, $m = 1, \dots, n$,
such that
$\bigcap_m g_mH_{\alpha_m}g_m^{-1}$ is not contained in a
parabolic $P\in \PP$. Then there exists a vertex
$z \in G$
such that the ball of radius
$C$
in $\Gamma$
intersects every coset $g_mH_{\alpha_m}$.
Further, for any $i \in \{1, \dots , n\}$, there are only finitely many double cosets of the form
$H_i g_i H_{\alpha_i}$
such that
$H_i \cap \bigcap_i g_i H_{\alpha_i}g_i^{-1}$
is not contained in a parabolic $P\in \PP$.
Let $G$ be a relatively hyperbolic group, and let $H$ be a relatively quasiconvex subgroup.
Then $H$ has finite relative algebraic height.\end{theorem}
This allows us to give an example of geometric height in our setting.
\begin{prop}\label{prop;rh_or_gh}
Let $(G, \calP)$ be a relatively hyperbolic group, and
$(G,d)$ a relative word metric $d$ (\ie a word
metric over a generating set that is the
union of a finite set and of a set of conjugacy representatives of the
elements of $\calP$, and hence, in general, not a finite generating
set). Let $H$ be a relatively quasiconvex subgroup. Then, $H$ has
finite geometric height for $d$.
\end{prop}
This just a rephrasing of Hruska and Wise's
result Theorem \ref{relht}. The proof is similar to
that in the hyperbolic groups case, using for
instance cones instead of balls.
\subsection{Mapping class groups}\label{mcg}
Another source of examples arise from convex-cocompact subgroups of
mapping class groups, and of $\Out(F_n)$ for a free group $F_n$. We establish
finiteness of both algebraic and geometric height
of convex cocompact subgroups of mapping class groups in
this subsubsection. In the following $S$ will be a closed
oriented surface of genus greater than $2$, and
$\Teich(S)$ and $\rCC(S)$ will denote respectively the Teichmüller space and Curve Complex of $S$.
\begin{defn}[\cite{farb-mosher}]
A finitely generated subgroup $H$ of the mapping class group
$\Mod(S)$ for a surface $S$ (with or without punctures) is
$\sigma$-convex cocompact if for some (any) $x\in \Teich(S)$, the
Teichmüller space of $S$, the orbit $Hx \subset \Teich(S)$ is
$\sigma$-quasiconvex with respect to the Teichmüller metric.
\end{defn}
Kent-Leininger \cite{kl} and Hamenstädt \cite{ham-cc} prove the following:
\begin{theorem}\label{theo;klh}
A finitely generated subgroup $H$ of the mapping class group
$\Mod(S)$ is convex cocompact if and only if for some (any) $x\in
\rCC(S)$, the curve complex of~$S$, the orbit $Hx \subset \rCC(S)$ is
qi-embedded in $\rCC(S)$.\label{qi-coco}
\end{theorem}
One important ingredient in Kent-Leininger's proof of Theorem
\ref{qi-coco} is a lifting of the limit set of $H$ in $\partial \rCC(S)$
(the boundary of the curve complex) to $\partial \Teich(S)$ (the boundary
of Teichmüller space). What is important here is that $\Teich(S)$ is a
proper metric space unlike $\rCC(S)$. Further, they show using a
theorem of Masur \cite{masur}, that any two Teichmüller geodesics
converging to a point on the limit set~$\Lambda_H$ (in $\partial
\Teich(S)$) of a convex cocompact subgroup $H$ are asymptotic. An
alternate proof is given by Hamenstädt in \cite{ham-gd}. With these
ingredients in place, the proof of Theorem \ref{alght-mcg0} below is
an exact replica of the proof of Theorem \ref{alght} above:
\begin{theorem}[Height from the Teichmüller metric]\label{alght-mcg0}
Let $G$ be the mapping class group of a surface $S$, and
$\Teich(S)$ the corresponding Teichmüller space with the
Teichmüller metric, and with a
base point $z_0$. Then for
$\sigma \geq 0$, there exists $C\geq 0$, and $D\geq 0$ such that the
following holds.
Let $H_1, \dots , H_n$ be a finite collection of $\sigma$-convex
cocompact subgroups of $G$. Suppose that there exist distinct
cosets $\{ g_mH_{\alpha_m}\}$ with $\alpha_m \in \{1, \dots ,
n\}$, $m = 1, \dots, n$, such that, for some $\Delta$,
$\bigcap_m (g_mH_{\alpha_m})^{+\Delta}$ is larger than $\max\{10\Delta,
D\}$. Then there exists a point $z \in \Teich(S)$ such that the
ball of radius $C$ in $\Teich(S)$ intersects every image
of $z_0$ by a coset $g_mH_{\alpha_m} z_0$.
Further, for any $i \in \{1, \dots , n\}$, there are only
finitely many double cosets of the form $H_i g_i H_{\alpha_i}$
such that $H_i \cap \bigcap_i g_i H_{\alpha_i}g_i^{-1}$
is infinite.
The collection $\{ H_1, \dots , H_n\}$ has finite algebraic height.
\end{theorem}
A more geometric strengthening of Theorem \ref{alght-mcg0} can be
obtained as follows using recent work of Durham and Taylor
\cite{DuTa}, who have given an intrinsic quasi-convexity
interpretation of convex cocompactness, by proving that convex
cocompact subgroups of mapping class groups are stable: in
a word metric, they are
undistorted, and quasi geodesics with end points in the subgroup remain close
to each other \cite{DuTa}.
\begin{theorem}[Height from a word metric]\label{alght-mcg}
Let $G$ be the mapping class group of a surface $S$ and $d$ the word
metric with respect to a finite generating set.
Then for $\sigma \geq 0$, and any subgroup $H$ that is
$\sigma$-convex cocompact, the group $H$ has finite geometric height in $(G,d)$.
Moreover, any $\sigma$-convex cocompact subgroup $H$ has finite
geometric height in $(G,d_1)$, where
$d_1$ is the word metric with respect to any (not
necessarily finite) generating set.
\end{theorem}
\begin{proof}
Assume that the theorem is false: there exists $\sigma$ such
that for all $k$, and all~$D$, there exists a
$\sigma$-convex
cocompact subgroup $H$, with a collection of distinct cosets $\{ g_mH,
m=0, \dots, k\}$ (with $g_0=1$), satisfying the property that $\bigcap_m (g_mH)^{+\Delta}$
has diameter larger than
$\max\{10\Delta, D\}$.
Let $a,b$ be two points in $\bigcap_m (g_mH)^{+\Delta}$ such that
$d(a,b) \geq \max\{10\Delta, D\}$. For each~$i$, let $a_i, b_i$ in
$g_i H$ be at distance at most $\Delta$ from $a$ and $b$
respectively. Consider $\gamma_i$ geodesics in $H$ from $g_i^{-1}
a_i$ to $g_i^{-1} b_i$. Consider also $a'_i$ and $b'_i$-nearest
point projections of $a_0$ and $b_0$ on $g_i \gamma_i$. Finally,
denote by $g_i \gamma'_i$ the subpath of $g_i \gamma_i$
between $a'_i$ and $b'_i$
By \cite[Prop.\,5.7]{DuTa}, $H$ is quasiconvex in $G$ (for a fixed chosen
word metric), and
for each $i$, $g_i \gamma_i$ is a
$f(\sigma)$-quasi-geodesic (for some function $f$).
We thus obtain from $a_0$ to $b_0$ a family of $k+1$ paths, namely
$\gamma_0$ and (one for all~$i$), the concatenation $ \eta_i = [a_0, a'_i] \cdot g_i \gamma'_i \cdot [b'_i,b_0]$. For $D$ large enough, the paths $\eta_i$ are $2f(\sigma)$-quasigeodesics in $G$.
Stability of $H$ (\cite[Th.\,1.1]{DuTa}) implies
that, there exists $R(\sigma)$ such that in $G$, the paths remain
at mutual Hausdorff distance at most
$R(\sigma)$. This is thus also true in the
Teichmüller space by the orbit map. Hence it follows that all the
subpaths $g_i \gamma'_i$ are at distance at most $2 R(\sigma)$
from each other, but are disjoint, and all lie in a thick part of the
Teichmüller space, where the action is uniformly proper. This leads to
a contradiction.
Since the diameter of intersections can only go down if the
generating set is increased, the last statement follows.
\end{proof}
\subsection{$\Out(F_n)$} \label{outfn}
Following Dowdall-Taylor \cite{dt1}, we say that
a finitely generated subgroup $H$ of $\Out(F_n)$
is
$\sigma$-\emph{convex cocompact}
if
\begin{enumerate}
\item all non-trivial elements of $H$ are atoroidal and fully irreducible,
\item for
some (any)
$x\in
cv_n$, the (projectivized) Outer space for $F_n$, the orbit $Hx \subset cv_n$
is $\sigma$-quasiconvex with respect to the Lipschitz metric.
\end{enumerate}
\begin{rmark}
The above lemma, while not explicit in \cite{dt1}, is implicit in Section 1.2 of that paper.
Also, a word about the metric on $cv_n$ is in order. The statements in \cite{dt1} are made with respect to the {\em unsymmetrized} metric on outer space. However, convex cocompact subgroups have orbits lying in the thick part; and hence the unsymmetrized and symmetrized metrics are quasi-isometric to each other. We assume henceforth, therefore, that we are working with the symmetrized metric, to which the conclusions of \cite{dt1} apply via this quasi-isometry.
\end{rmark}
The following theorem gives a characterization of convex
cocompact subgroups in this context and is the analog of
Theorem \ref{qi-coco}.
\begin{theorem}[\cite{dt1}] \label{theo;dt14}
Let $H $ be a finitely generated subgroup of $\Out(F_n)$ all whose
non-trivial elements are atoroidal and fully irreducible. Then
$H$ is convex cocompact
if and only if for
some (any)
$x\in
\FF_n$ (the free factor complex of $F_n$), the orbit $Hx \subset \FF_n$
is qi-embedded in $\FF_n$.\label{qi-coco-out}\end{theorem}
Dowdall and Taylor also show \cite[Th.\,4.1]{dt1} that any two
quasi-geodesics in $cv_n$
converging to the same point $p$ on the limit set $\Lambda_H$ (in
$\partial cv_n$) of a convex cocompact subgroup $H$
are asymptotic. More precisely, given $\lambda, \mu$ and $p \in
\Lambda_H$ there exists $C_0(= C_0(\lambda, \mu, p))$ such that any
two
$(\lambda, \mu)$-quasi-geodesics in $cv_n$ converging to
$p$ are asymptotically $C_0$-close.
As observed before in the context of Theorem \ref{alght-mcg0},
this is adequate for the proof of Theorem \ref{alght-mcg0} to go through:
\begin{theorem} \label{alght-out}
Let $G=\Out(F_n)$, and $cv_n$ the Outer space for $G$ with
a base point~$z_0$.
Then for $\sigma \geq 0$, there exists $C\geq 0$
such that the following holds.
Let $H_1, \dots , H_n$
be a finite collection of $\sigma$-convex cocompact subgroups of $G$.
Suppose that there exist distinct cosets $\{
g_mH_{\alpha_m}\}$ with $\alpha_m \in \{1, \dots ,
n\}$, $m = 1, \dots, n$,
such that
$\bigcap_m g_mH_{\alpha_m}g_m^{-1}$ is infinite. Then there exists a point
$z \in cv_n$
such that the ball of radius
$C$
in $cv_n$
intersects every image of $z_0$ by a coset
$g_mH_{\alpha_m} z_0$.
Further, for any $i \in \{1, \dots , n\}$, there are
only finitely many double cosets of the form
$H_i g_i H_{\alpha_i}$
such that
$H_i \cap \bigcap_i g_i H_{\alpha_i}g_i^{-1}$
is infinite.
The collection $\{ H_1, \dots , H_n\}$ has finite algebraic height.
\end{theorem}
Since an analog of the stability result of \cite{DuTa}
in the context of $\Out(F_n)$ is missing at the moment, we
cannot quite get an analog
of Theorem \ref{alght-mcg}.
\Subsection{Algebraic and geometric qi-intersection
property: Examples}
In the proposition below
we shall put parentheses
around (geometric) to indicate that the statement holds for
both the qi-intersection property
as well as the
geometric qi-intersection property.
\pagebreak[2]
\skpt
\begin{prop}\label{prop;satisfiesqiip}
\begin{enumerate}
\item Let $H$ be a quasiconvex subgroup of a hyperbolic
group $G$, endowed
with a locally finite word metric. Then,
$\{H\}$ satisfies the uniform (geometric) qi-intersection property.
\item Let $H$ be a relatively quasiconvex subgroup of a relatively
hyperbolic group $(G, \calP)$. Let $\calP_0$ be a set of conjugacy
representatives of groups in $\calP$, and $d$ a word metric on $G$
over a generating set $S= S_0\cup \calP_0$, where $S_0$ is
finite. Then $\{H\}$ satisfies the uniform (geometric) qi-intersection property
with respect to $d$.
\item Let $H$ be a convex-cocompact subgroup of the Mapping Class
Group $\Mod(\Sigma)$ of an oriented closed surface $\Sigma$ of genus
$\geq 2$. If $d$ is a word metric on $\Mod(\Sigma)$ that makes it
quasi-isometric to the curve complex of $\Sigma$, then
$H$ satisfies the uniform (geometric) qi-intersection
property
with respect to $d$.
\item Let $H$ be a convex-cocompact subgroup of $\Out(F_n)$ for
some $n\!\geq\!2$. If $d$ is a word metric on $\Out(F_n)$ that makes it
quasi-isometric to the free factor complex of~$F_n$,
then~$H$ satisfies the uniform (geometric)
qi-intersection property
with respect to~$d$.
\end{enumerate}
\end{prop}
\begin{proof}
All four cases have similar proofs. Consider the first point.\vspace*{-6pt}\enlargethispage{.5\baselineskip}%
\subsubsection*{Case 1: $G$ hyperbolic, $H$ quasiconvex}
Let $h$ be the height of $H$ (which is finite by Theorem
\ref{alght}): every $h+1$-fold intersection of conjugates of $H$ is finite, but some $h$-fold intersection is infinite.
\begin{itemize}
\item
The first conditions of Definition \ref{qiintn} and
Definition \ref{geoqiintn} follow from this finiteness and
Proposition \ref{prop;hyp_qc_have_fgh}
and Theorem \ref{short-intn}.
\item
The second condition of Definition \ref{geoqiintn}
follows from Proposition \ref{prop;projections}.
\item
We prove the second condition of Definition \ref{qiintn} (on mutual coboundedness of elements of $\CC\HH_i$) iteratively.
\end{itemize}
By Theorem \ref{short-intn}, there exists $C_h$ such that two elements of $\CC\HH_h$ are $C_h$-quasiconvex in $(G,d)$. Let $D>0$. If $A$ and $B$ are two distinct such elements such that the projection of $A$ on $B$ has diameter greater than $D$, then there are $D/C_h$ pairs of elements $(a_i, b_i)$ in $A\times B$, such that $a_i^{-1}b_i $ are elements of length at most $ 20\delta C_h$. Choose $N_0$ larger than the cardinality of finite subgroups of $G$. By a standard pigeon hole argument, if $D$ is large enough, there are $N_0$ such pairs for which $a_i^{-1}b_i $ take the same value. It follows that there are two essentially distinct conjugates of elements of $\HH_h$ that intersect on a subset of at least $N_0$ elements, hence on an infinite subgroup. This contradicts the definition of height, and it follows that $D$ is bounded, and elements of $\CC\HH_h$ are mutually cobounded.
We continue by descending induction. Assume that the second property of Definition \ref{qiintn} is established for
$\CC\HH_{i+1}$. By Proposition \ref{prop;criterion} it follows that $(G,d_{i+1})$ is hyperbolic.
Let $\delta_{i+1}$ be its hyperbolicity constant.
By Proposition \ref{cobpersists}, there exists~$C_i$ such that two elements of $\CC\HH_{i}$ are $C_i$-quasiconvex in $(G,d_{i+1})$.
Again take $A$ and $B$ two distinct elements of $\CC\HH_{i}$ such
that the projection of~$A$ on~$B$ has diameter greater than $D>1000\delta_{i+1}$ for
$d_{i+1}$. Then there are at least $D/C_i$ pairs of elements $(a_i,b_i)$ in $A\times
B$, such that $a_i^{-1}b_i $ is an element of length at most $
20\delta_{i+1} C_h$ for the metric $d_{i+1}$,
and for all $i$ there exists $i'$ such that the segments
$[a_i,b_i], [a_{i'}, b_{i'}]$ are $(200\delta_{i+1})$-far from one
another. Apply the Proposition \ref{prop;unfolding_qc} to each
geodesic $[a_i, b_i]$ to find quasi-geodesics $q_i$ from $a_i$ to
$b_i$ in $(G,d)$ (this can also be done by Lemma \ref{ea-strong}). We
know that in $(G,d)$, $A$ and $B$ are quasiconvex (for the constant~$C_h$). By their definition, and by hyperbolicity, the paths $q_i$
end at bounded distance of a shortest-point projection of $a_i$ to $B$
(for $d$).
Therefore, since $(G,d)$ is
hyperbolic, and since the $q_i$ are far from one another for $d$, it
follows that the $q_i$ are actually short for the metric $d$ (shorter than $(200\delta
C_h)$). Since there are $D/C_i$ pairs of elements $(a_i,b_i)$, by
the pigeon hole argument, there is an element $g_0$ (of length at most
$(200\delta
C_h)$ in the metric $d$) such that for $D/(C_i\times
B_{G,d}(200\delta C_h) )$ such pairs, the difference $a^{-1} b$ equals $g_0$. If $D$
is large enough, $D/(C_i \times B_{G,d}(200\delta
C_h) )$ is larger than the cardinality of the finite order elements of
$G$. It follows that the two essentially distinct conjugates of elements
of $\HH_i$, corresponding to the cosets~$A$ and~$B$, intersect on a
set of size larger than any finite subgroup of $G$ (and of diameter
larger than~$3$ in~$d_{i+1}$). Thus the
intersection is an infinite subgroup of $G$. This subgroup is necessarily among the conjugates of some $\HH_j$ for $j\geq i+1$, but therefore must have diameter $2$ in the metric $d_{i+1}$.\vspace*{-5pt}\vskip0pt\enlargethispage{.7\baselineskip}%
\subsubsection*{Case 2: $G$ relatively hyperbolic, $H$
relatively quasiconvex}
The geometric height of $H$ for the relative metric is
finite, by Proposition \ref{prop;rh_or_gh}. Let $h$ be
its value.
The first points of Definition \ref{qiintn} follows from this finiteness and
Theorem \ref{hruska-intn}.
The second point has a similar proof as the first case, except that the pigeon hole argument needs to be made precise because the relative metric $(G,d)$ is not locally finite.
Let $D>0$. If $A$ and $B$ are two
distinct elements of $\CC\HH_h$ such that the projection of $A$ on $B$ has diameter
greater than $D$, then there are $D/C_H$ pairs of elements $(a_i,
b_i)$ in $A\times B$, such that $a_i^{-1}b_i $ are elements
of length at most $ 20\delta C_h$. Moreover, if $D>100\delta C_h$,
for each $[a_i,b_i]$, there is $[a_j, b_j]$ such that both segments
are short (for~$d$) and are
at distance at least $(50\delta C_h)$ from each other. It
follows that, in the Cone-off Cayley graph of $G$, the maximal angle
of $[a_i, b_i]$ at the cone vertices is uniformly bounded
by $(100\delta C_h) + 2(2C_h+5\delta)$. Indeed, consider
$\alpha$ and $\beta$
quasi-geodesic paths in $A$ and $B$ respectively, from
$a_i$ to $a_j$ and from $b_i$ to $b_j$. By hyperbolicity and
quasi-geodesy, at distance $30\delta C_h$ from $a_i$ and
$b_i$, there is a path of length $2(2C_h+5\delta)$ joining~$\alpha$ to~$\beta$. Being too short, this path cannot possibly intersect $[a_i,
b_i]$. There is thus a path from $a_i$ to $b_i$ of length at most
$2\times (30\delta C_h) + 2(2C_h+5\delta)$ that does not intersect
$[a_i, b_i]$ outside its end points. It follows indeed that the
maximal angle of $[a_i, b_i]$ is at most $2\times (30\delta C_h) +
2(2C_h+5\delta)+ 20\delta C_h$.
From this bound on angles, we may use the fact that the angular metric
at each cone point is locally finite (by definition of relative
hyperbolicity) and the bound on the length in the
metric $d$, to get that all the elements $a_i^{-1}b_i $ are in a
finite set, independent of $D$. We can now use the pigeon hole argument, as in the hyperbolic case, and conclude similarly that $D$ is bounded.
The rest of the argument is also by descending induction. Assume that
the second property of Definition \ref{qiintn} is established for
$\CC\HH_{i+1}$.
We proceed in a very similar way as in the hyperbolic case, with the
difference is that, after establishing that the paths~$q_i$ are small
for the metric $d$, one needs to check that their angles at cone
points are bounded, which is done by the argument we just used. We
provide the details~now.
By Proposition \ref{prop;criterion} it follows that $(G,d_{i+1})$ is hyperbolic.
Let $\delta_{i+1}$ be its hyperbolicity constant.
By Proposition \ref{cobpersists}, there exists $C_i$ such that two elements of $\CC\HH_{i}$ are $C_i$-quasiconvex in $(G,d_{i+1})$.
Take $A$ and $B$ two distinct elements of $\CC\HH_{i}$ such
that the projection of $A$ on $B$ has diameter greater than
some constant $D$ for
$d_{i+1}$. Take a quasigeodesic in the projection of $A$ on $B$, of
length $D$. Then there are at least $D/C_i$ pairs of elements $(a_i,b_i)$ in $A\times
B$, with $b_i$ on that quasigeodesic, and such that $a_i^{-1}b_i $ is an element of length at most $
20\delta_{i+1} C_h$ for the metric $d_{i+1}$,
and for all $i$ there exists $i'$ such that the segments
$[a_i,b_i], [a_{i'}, b_{i'}]$ are $(200\delta_{i+1})$-far from one
another. Apply the Proposition \ref{prop;unfolding_qc} (or
alternatively \ref{ea-strong}) to each
geodesic $[a_i, b_i]$ to find quasi-geodesics $q_i$ from $a_i$ to~$b_i$ in $(G,d)$. We
know that in $(G,d)$, $A$ and $B$ are quasiconvex (for the constant~$C_h$). By their definition, and by hyperbolicity, the paths $q_i$
end at bounded distance of a shortest-point projection of $a_i$ to $B$
(for $d$).
Therefore, since $(G,d)$ is
hyperbolic, and since the $q_i$ are far from one another for $d$, it
follows that the $q_i$ are actually short for the metric $d$ (shorter than $(200\delta
C_h)$).
By the argument used at the initial step of the descending induction, we also have an uniform upper bound on the
maximal angle of these paths, and therefore on the number of elements
of $G$ that label one of the paths~$q_i$.
Since there are $D/C_i$ pairs of elements $(a_i,b_i)$, if $D$ is
large enough, by
the pigeon hole argument, there is an element $g_0$ (of length at most
$(200\delta
C_h)$ in the metric~$d$), and a pair $(a_{i_0}, b_{i_0})$, such that
$a^{-1}_i b_i = g_0$ and such that
for $1000\delta_{i+1} C_i$
other such pairs $(a_j, b_j)$, the difference $a^{-1}_j b_j$ is also
equal to $g_0$. The intersection of two essentially distinct conjugates of elements
of $\HH_i$, corresponding to the cosets $A$ and $B$, thus contains
$a_{i_0}^{-1} a_j$ for all those indices $j$. There are indices $j$ for which $a_{i_0}^{-1} a_j$ labels
a quasi-geodesic paths in $A$ of length at least
$1000\delta_{i+1} C_i $. Such an element is either loxodromic,
or elliptic with fixed point at the midpoint $[a_{i_0}, a_j]$. But
if all of them are elliptic,
for two indices $j_1, j_2$, we get two different fixed points, hence the
product of the elements $a_{i_0}^{-1} a_{j_1} a_{i_0}^{-1} a_{j_2}$ is loxodromic.
This element is in the intersection of conjugates
of elements of $\HH_i$, thus is in a subgroup among the conjugates
of some $\HH_j$ for $j\geq i+1$, but therefore must have diameter $2$
in the metric $d_{i+1}$, and cannot contain loxodromic elements. This
is thus a contradiction.
\subsubsection*{Cases 3 and 4: $G= \Mod(\Sigma)$ or
$\Out(F_n)$, $H$ convex cocompact}
Consider the Teichmüller metric on Teichmüller space
$(\Teich(\Sigma),d_T)$
and the
(symmetrization of~the) Lipschitz metric on Outer space
$(cv_n,d_S)$
respectively for
$\Mod(\Sigma)$
and $\Out(F_n)$.
Though $\Teich(\Sigma)$ and $cv_n$ are
non-hyperbolic, they
are proper metric spaces.
For the mapping class group $\Mod(\Sigma)$, the curve
complex $(\rCC\Sigma), d)$ is hyperbolic and quasi-isometric to $(\Mod(\Sigma),d)$, where $d$ is
the word-metric on $\Mod(\Sigma)$ obtained by taking as
generating set a finite generating
set of $\Mod(\Sigma)$ along with {\it all} elements of
certain sub-mapping class groups (see \cite{masur-minsky}).
Similarly for $\Out(F_n)$, the free factor complex
$(\FF_n, d) $
is hyperbolic, and is quasi-isometric to $(\Out(F_n),d)$
for a certain word metric over an infinite generating
set (\cite{BF-ff}).
This establishes that the hypotheses in the statements of
Cases 3 and 4 are not vacuous.
Recall that if a subgroup $H$ of $\Mod(\Sigma)$ or $\Out(F_n)$
is $C$-convex co-compact, then by Theorem \ref{theo;klh} (and
\ref{theo;dt14}) the orbit of a base point
in $\rCC\Sigma)$ (or $\FF_n$) is a quasi-isometric image
of the orbit of a base point in Teichmüller space.
Finiteness of height of convex cocompact subgroups
follows from Theorems \ref{alght-mcg} and~\ref{alght-out}
for
$G= \Mod(\Sigma)$ and $\Out(F_n)$ respectively.
The first condition of Definition~\ref{qiintn} now follows from Theorems
\ref{short-intn-coco}.
We now proceed with proving the second condition of Definition \ref{qiintn}. We first remark that, given $C$, there exists $\Delta, C'$ such that if $A,B$ are cosets of $C$-convex
co-compact subgroups, and if $a_1, a_2 \in A$, $b_1,
b_2\in B$ are such that, in $\rCC\Sigma)$, $d(a_1, b_1)$
and $d(a_2, b_2) $ are at most $10C\delta$ and that $d(a_1, a_2)$
and $d(b_1,b_2)$ are larger than $\Delta$ then, $d_T(
a_i, b_i ) \leq C' $ for both $i=1,2$. Indeed, by
definition of convex cocompactness, the segment $[a_1,
a_2 ]$ in Teichmüller space maps on a parametrized
quasi-geodesic in the curve complex. A result of
Dowdall Duchin and Masur ensures that
Teichmüller geodesics that make progress in the curve
complex, are contracting in Teichmüller space
\cite[Th.\,A]{DDM} (see the formulation done and proved in
\cite[Prop.\,3.6]{DH}). Thus the
segment $[a_1,
a_2 ]$ is contracting in Teichmüller space: any Teichmüller
geodesic whose projection in the curve complex
fellow-travels that of $[a_1,
a_2 ]$ has to be uniformly close to $[a_1,
a_2 ]$. Applying that to
the segment $[b_1, b_2]$, it follows
that it must remain at bounded distance (for Teichmüller
distance) from $[a_1, a_2]$, as demanded.
A
similar statement is valid for $\Out(F_n)$ with the
objects that we introduced, it suffice to use \cite[Prop.\,4.17]{DH}, an arrangement of Dowdall-Taylor's result
\cite{dt1}, in place of the Dowdall-Duchin-Masur criterion.
With this estimate, one can easily adapt the proof of the
first case to get the result.
\end{proof}
\section{From quasiconvexity to graded relative hyperbolicity}
Recall that we defined graded
geometric relative hyperbolicity in
Definition \ref{ggrh}.
\Subsection{Ensuring geometric graded relative hyperbolicity}
\begin{prop} \label{ghqiimpliesggrh-a}
Let $G$ be a group, $d$ a word metric on $G$
with respect to some (not necessarily finite)
generating set, such that $(G,d)$ is
hyperbolic.
Let $H$ be a subgroup of $G$.
If $\{H\}$ has finite
geometric height for $d$ and has the uniform qi-intersection
property, then $(G,\{H\},d)$ has graded
geometric relative hyperbolicity.
\end{prop}
\begin{proof}
As in Definition \ref{qiintn}, $\HH_n$ denotes the collection of
intersections of $n$
essentially distinct conjugates of $H$. Let
$(\HH_n)_0$ denote a set of conjugacy
representatives of $(\HH_n)$ that are $C_1$-quasiconvex, and let
$\CC\HH_n$ denote the collection of cosets of elements of $(\HH_n)_0$.
Let $d_n$ be the metric on $X=(G,d)$ after electrifying the elements of
$\CC\HH_n$.
By Definition \ref{qiintn}
and Remark \ref{rem:cobdd}, for all $n$, all elements of $\CC\HH_n$ and
of $\CC\HH_{n+1}$ are
$C_1$-quasiconvex in $(G,d)$. Therefore, by Proposition
\ref{cobpersists}, all elements of $\CC\HH_n$ are
$C'_1$-quasiconvex in $(G,d_{n+1})$ for
some $C'_1$ depending on the hyperbolicity of $d$, and on~$C_1$.
By Definition \ref{qiintn}, $\CC\HH_n$ is mutually cobounded in
the metric $d_{n+1}$. Proposition
\ref{prop;criterion} now shows that the horoballification of $(G,d_{n+1})$
over $\CC\HH_{n}$ is hyperbolic, for all~$n$. Proposition
\ref{prop;from_horo_to_he} then guarantees that $\CC\HH_{n}$ is
coarsely hyperbolically embedded in $(G,d_{n+1})$. Since $H$ is assumed to
have finite
geometric height, $(G,\{H\},d)$ has
graded geometric relative hyperbolicity.
\end{proof}
\Subsection{Graded relative hyperbolicity for quasiconvex subgroups}
\begin{prop}\label{qcimpliesgrh}
Let $H$ be a quasiconvex subgroup of a hyperbolic group $G$, with a
word metric $d$ (with respect to a finite generating set). Then the
pair $(G,\{H\})$ has
graded geometric
relative hyperbolicity, and graded relative hyperbolicity.
\end{prop}
\begin{proof}
For the word metric $d$ with respect to a finite generating set,
graded geometric relative hyperbolicity agrees with the notion of graded
relative hyperbolicity (Definition \ref{grh}).
By Theorem \ref{alght}, $H$ has finite height. By Proposition
\ref{prop;satisfiesqiip} it satisfies the uniform qi-intersection
property \ref{qiintn}. Therefore, by Proposition
\ref{ghqiimpliesggrh-a}, the pair $(G,\{H\})$ has
graded relative
hyperbolicity.
Finally, note that since the word metric we use is locally finite, and all $i$-fold intersections are quasiconvex, graded relative
hyperbolicity follows.
\end{proof}
\begin{prop} \label{relqcimpliesgrh}
Let $(G,\PP)$ be a finitely generated relatively
hyperbolic group. Let~$H$ be a relatively quasiconvex
subgroup. Let $S$ be a finite relative generating set of~$G$ (relative to $\PP$) and let
$d$ be the word metric with respect to $S \cup \PP$. Then $(G,\{H\},d)$ has
graded relative hyperbolicity as well as
graded geometric relative hyperbolicity.
\end{prop}
\begin{proof}
The proof is similar to that of Proposition \ref{qcimpliesgrh}. By
Theorem \ref{relht}, $H$ has finite relative height, hence it has
finite
geometric
height for the relative metric (see Example
\ref{prop;rh_or_gh}).
Next, by Proposition \ref{prop;satisfiesqiip}, $H$ satisfies the uniform
qi-intersection property for a relative metric, and
graded geometric relative hyperbolicity follows from
Proposition~\ref{ghqiimpliesggrh-a}.
Again, since $G$ has a word metric with respect to a
finite relative generating set, and~$H$ and all $i$-fold
intersections are relatively
quasiconvex as well, the above argument furnishes graded
relative hyperbolicity as well.
\end{proof}
Similarly, replacing the use of Theorem
\ref{alght} by Theorems \ref{alght-mcg0} and \ref{alght-out},
one obtains the following.
\begin{prop}\label{cocoimpliesgrh0} Let $G$ be the mapping
class group $\Mod(S)$ (respectively $\Out(F_n)$). Let $d$
be a word metric
on $G$ making it quasi-isometric to the curve complex
$\rCC(S)$ (respectively the free factor complex $\FF_n$).
Let $H$ be a convex cocompact subgroup of~$G$. Then $(G,\{H\},d)$ has
graded relative hyperbolicity.
\end{prop}
Again, replacing the use of Theorem
\ref{alght} by Theorem \ref{alght-mcg}, we obtain:
\begin{prop}\label{cocoimpliesgrh} Let $G$ be the mapping
class group $\Mod(S)$. Let $d$ be a word metric
on $G$ making it quasi-isometric to the curve complex
$\rCC(S)$.
Let $H$ be a convex cocompact subgroup of $G$. Then $(G,\{H\},d)$ has
graded geometric relative hyperbolicity.
\end{prop}
\begin{remark} Since we do not have an exact (geometric)
analog of Theorem \ref{alght-mcg} for $\Out(F_n$) (more
precisely an analog of the stability result of
\cite{DuTa}) as of now, we have to content ourselves with
the slightly weaker Proposition \ref{cocoimpliesgrh0} for $\Out(F_n$).
\end{remark}
\section{From graded relative hyperbolicity to quasiconvexity}
\Subsection{A sufficient condition}
\begin{prop}\label{ggrhtoqc}
Let $G$ be a group and $d$ a hyperbolic word metric with respect to a (not
necessarily finite) generating set.
Let $H$ be a subgroup such that $(G,\{H\},d)$ has
graded geometric
relative hyperbolicity.
Then $H$ is quasiconvex in $(G,d)$.
\end{prop}
\begin{proof} Assume $(G,\{H\},d)$ has
graded geometric relative
hyperbolicity as in Definition \ref{ggrh}. Then $H$ has finite geometric height in
$(G,d)$. Let $k$ be this height. Thus, $\HH_{k+1}$ is a collection
of uniformly bounded subsets, and
$d_{k+1} $ is quasi-isometric to $d$. It follows
that $(G,d_{k+1})$ is hyperbolic.
Further, by Definition \ref{ggrh},
$\HH_k$ is hyperbolically embedded in
$(G,d_{k+1})$. This means in particular that the
electrification $(G,d_{k+1})^\rmel_{\HH_k} $
is hyperbolic. Since $(G,d_k)$ is quasi-isometric to
$(G,d_{k+1})^\rmel_{\HH_k} $ (being
the restriction of the metric on $G$)
it follows that $(G,d_k)$ is hyperbolic as well. Further,
by Corollary \ref{cor-retraction} the elements
of $\HH_k$, are uniformly quasiconvex in $(G,d_{k+1})$.
We now argue by descending induction on $i$.
\subsubsection*{The inductive hypothesis for $(i+1)$}
We assume that $d_{i+1}$ is a hyperbolic metric on~$G$,
and that there is a constant $c_{i+1}$ such that, for
all $j\geq 1$
the elements of $ \HH_{i+j}$
are uniformly $c_{i+1}$-quasiconvex in $(G,d)$.
We assume the inductive hypothesis for $i+1$ (\ie as
stated), and we now prove it for $i$.
Of course, we also assume, as in the statement of the proposition,
that
$\HH_{i}$ is coarsely hyperbolically embedded in
$(G,d_{i+1})$. Hence $d_i$ is a hyperbolic
metric on $G$.
We will now check that the assumptions of Proposition
\ref{prop;unfolding_qc}
are satisfied for $(X,d) = (G, d_{i+1})$, $\YY =
\HH_{i+1}$, and $H_{i,\ell}$
arbitrary in $\HH_i$.
Elements of $\HH_{i}$
in $(G,d_{i+1})$ are uniformly quasiconvex in $(G,d_{i+1})$: this
follows from Corollary \ref{cor-retraction}. We will write
$C_i$ for their quasiconvexity constant.
A second step is to check that, for some uniform $\Delta_0$ and
$\epsilon$, for all $\Delta>\Delta_0$, when an element
$H_{i,\ell}$ of $\HH_{i}$
$(\Delta,\epsilon)$-meets an item of $\HH_{i+1}$, then
$H^{+\epsilon \Delta}$ contains
a quasigeodesic between the meeting points in $H$. Thus, fix
$\epsilon <1/100$, and take $\Delta_0 $
larger than $20$ times the thickening constants for the definition
of elements in $\HH_i$ (which is possible by finiteness
of number of orbits of $i$-fold intersections).
Assume
$H_{i,\ell}$ $(\Delta, \epsilon)$-meets $Y \in \HH_{i+1}$.
Then, by definition of $i$-fold intersections
\ref{def:ifoldinter}, and Proposition \ref{geointnsstable},
either the pair of meeting points is in an item of
$\HH_{i+1}$ inside $H_{i,\ell} $, or
$Y\subset H_{i,\ell}$.
In both cases, by the inductive assumption,
there is a path in $H_{i,\ell}^{+\epsilon \Delta}$ between the meeting
points in~$H_{i,\ell}$ that is a quasigeodesic for
$d$. Hence the second assumption
of Proposition \ref{prop;unfolding_qc} is satisfied.
We can thus conclude by that proposition that $H_{i,\ell}$ is quasiconvex in $(G,d)$ for a
uniform constant, and therefore the inductive assumption holds for $i$.
By induction it is then true for $i=0$, hence the first
statement of the proposition holds, \ie quasiconvexity follows from
graded geometric relative hyperbolicity.
\end{proof}
We shall deduce various consequences of Proposition
\ref{ggrhtoqc} below. However, before we proceed, we need
the following observation
since we are dealing with spaces/graphs that are not necessarily proper.
\begin{obs} \label{qctoqi} Let $X$ be a (not necessarily
proper) hyperbolic graph. For all $C_0\geq 0$, there
exists
$C_1 \geq 0$ such that the following holds:
Let $H$ be a hyperbolic group acting uniformly properly
on $X$, \ie for all $D_0$ there exists $N$ such that for
any $x \in X$, any $D_0$ ball in $X$ contains at most $N$
orbit points of $Hx$.
Then a $C_0$-quasiconvex orbit of $H$ is
$(C_1,C_1)$-quasi-isometrically embedded in~$X$.
\end{obs}
Combining Proposition \ref{ggrhtoqc} with Observation
\ref{qctoqi} we obtain the following:
\begin{prop}\label{ggrhtoqi}
Let $G$ be a group and $d$ a hyperbolic
word metric with respect to a (not necessarily finite) generating set.
Let $H$ be a subgroup such that
\begin{enumerate}
\item $(G,\{H\},d)$ has graded geometric relative hyperbolicity,
\item The action of $H$ on $(G,d)$ is uniformly proper.
\end{enumerate}
Then $H$ is hyperbolic and
$H$ is qi-embedded in $(G,d)$.
\end{prop}
\begin{proof}
Quasi-convexity of $H$ in $(G,d)$ was established in Proposition \ref{ggrhtoqc}.
Qi\nobreakdash-em\-beddedness
of $H$ follows from Observation \ref{qctoqi}. Hyperbolicity of $H$ is
an immediate consequence.
\end{proof}
\subsection{The main theorem} We assemble the pieces now to
prove the following main theorem of the paper.
\begin{theorem}\label{hypcharzn} Let $(G,d)$ be one of the following:
\begin{enumerate}
\item $G$ a hyperbolic group and $d$ the word metric with
respect to a finite generating set $S$.
\item $G$ is finitely generated and hyperbolic relative
to $\PP$, $S$ a finite relative generating set, and
$d$
the word metric with respect to $S \cup \PP$.
\item $G$ is the mapping class group $\Mod(S)$ and $d$ the
metric obtained by electrifying the subgraphs
corresponding to
sub mapping class groups so that $(G,d)$ is
quasi-isometric to the curve complex $\rCC(S)$.
\item $G$ is $\Out(F_n)$ and $d$ the metric obtained by
electrifying the subgroups corresponding to
subgroups that stabilize proper free factors so that
$(G,d)$ is quasi-isometric to the free factor complex
$\FF_n$.
\end{enumerate}
Then (respectively)
\begin{enumerate}
\item $H$ is quasiconvex if and only if $(G,\{H\})$ has
graded geometric relative hyperbolicity,
\item $H$ is relatively quasiconvex if and only if
$(G,\{H\},d)$ has graded geometric relative
hyperbolicity,
\item $H$ is convex cocompact in $\Mod(S)$ if and only if
$(G,\{H\},d)$ has graded geometric relative
hyperbolicity and the action of
$H$ on the curve complex is uniformly proper,
\item $H$ is convex cocompact in $\Out(F_n)$ if and only
if $(G,\{H\},d)$ has graded geometric relative hyperbolicity
and the action of
$H$ on the free factor complex is uniformly proper.
\end{enumerate}
\end{theorem}
\begin{proof} The forward implications of quasiconvexity to graded
geometric relative hyperbolicity in the first 3 cases are proved by
Propositions \ref{qcimpliesgrh}, \ref{relqcimpliesgrh},
\ref{cocoimpliesgrh0} and \ref{cocoimpliesgrh}
and case 4 by Proposition \ref{cocoimpliesgrh0}. In cases
(3) and (4) properness of the action of $H$ on
the curve complex follows from convex cocompactness.
We now proceed with the reverse implications. Again, the
reverse implications of~(1) and (2) are direct
consequences
of Proposition \ref{ggrhtoqc}.
The proofs of the reverse implications of (3) and (4) are
similar. Proposition \ref{ggrhtoqi} proves that any
orbit of $H$ on
either the curve complex $\rCC(S)$ or the free factor
complex $\FF_n$ is qi-embedded. Convex cocompactness now
follows from
Theorems \ref{qi-coco} and \ref{qi-coco-out}.
\end{proof}
\subsection{Examples} We give a couple of examples below to show that
finiteness of geometric height does not necessarily follow from quasiconvexity.
\begin{example}
Let $G_1 = \pi_1(S)$ and $H =\langle h\rangle$ be
a cyclic subgroup corresponding to a simple closed
curve. Let $G_2 = H_1 \oplus H_2$, where each $H_i$ is isomorphic to $\mathbb{Z}$. Let $G
=G_1 \ast_{H=H_1} G_2$. Let $d$ be the metric obtained
on $G$ with respect to some finite generating
set along with all elements of $H_2$. Then $G_1$ is
quasiconvex in $(G,d)$, but~$G_1$ does not have finite
geometric height.
\end{example}
Note however, that the action of $G_1$ on $(G,d)$ is not
acylindrical. We now furnish another example to show that
graded geometric relative hyperbolicity does not
necessarily follow from quasiconvexity even if we assume
acylindricity.
\begin{example}
Let $G = \langle a_i, b_i: i \in
\natls, a_{2i}^{b_i} = a_{2i-1} \rangle$ and let $F$ be
the (free) subgroup
generated by $\{ a_i\}$. Then $F^{b_i}\cap F =
\langle a_{2i-1}\rangle$ for all $i$. Let $d$ be the word metric on
$G$ with respect to the generators
$a_i, b_i$. Then the action of $F$ on $(G,d)$ is
acylindrical and $F$ is quasiconvex. However there are
infinitely many double coset representatives
corresponding to $b_i$ such that $F^{b_i}\cap F $ is infinite.
\end{example}
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