HEIGHT, GRADED RELATIVE HYPERBOLICITY AND QUASICONVEXITY

. — 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 Out( F n ) . Résumé (Hauteur, hyperbolicité relative graduée, et quasiconvexité). — 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 exté-rieurs de groupe libre.


Introduction
It is well-known that quasiconvex subgroups of hyperbolic groups have finite height. In order to distinguish this notion from the notion of geometric height introduced later in this paper, we shall call the former algebraic height: Let G be a finitely generated group and H a subgroup. We say that a collection of conjugates {g i Hg −1 i }, i = 1, . . . , n are essentially distinct if the cosets {g i H} are distinct. We say that H has finite algebraic height if there exists n ∈ 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 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 , . . . , H n are quasiconvex), then H (or more generally the collection {H 1 , . . . , H n }) has finite algebraic height [GMRS98]. (See [Dah03,HW09] for generalizations to the context of relatively hyperbolic groups.) Swarup asked if the converse is true: Bes04]). -Let G be a hyperbolic group and H a finitely generated subgroup. If H has finite height, is H quasiconvex?
An example of an infinitely generated (and hence non-quasiconvex) malnormal subgroup of a finitely generated free group was obtained in [DM15] showing that the hypothesis that H is finitely generated cannot be relaxed. On the other hand, Bowditch shows in [Bow12] (see also [Mj08,Prop. 2.10]) the following positive result: Theorem 1.2 ( [Bow12]). -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.
One of the motivational points for this paper is to extend Theorem 1.2 to give a characterization of quasiconvex subgroups of hyperbolic groups in terms of a notion of graded relative hyperbolicity defined as follows: Definition 1.3. -Let G be a finitely generated group, d the word metric with respect to a finite generating set and H a subgroup. Let H i be the collection of intersections of i essentially distinct conjugates of H, let (H i ) 0 be a choice of conjugacy representatives, and let C H i be the set of cosets of elements of (H i ) 0 . Let d i be the metric on (G, d) obtained by electrifying (1) the elements of C H i . We say that G is graded relatively hyperbolic with respect to H (or equivalently that the pair (G, {H}) has graded relative hyperbolicity) if (1) H has algebraic height n for some n ∈ N, (1) The second author acknowledges the moderating influence of the first author on the more extremist terminology electrocution [Mj14,Mj10] J.É.P. -M., 2017, tome 4 (2) each element K of H i−1 has a finite relative generating set S K , relative to H ∩ H i (:= {H ∩ H i : H i ∈ H i }); further, the cardinality of the generating set S K is bounded by a number depending only on i (and not on K), (3) (G, d i ) is strongly hyperbolic relative to H i−1 , where each element K of H i−1 is equipped with the word metric coming from S K .
The following is the main theorem of this paper (see Theorem 6.4 for a more precise statement using the notion of graded geometric relative hyperbolicity defined later) providing a partial positive answer to Question 1.1 and a generalization of Theorem 1.2: Theorem 1.4. -Let (G, d) be one of the following: (1) G a hyperbolic group and d the word metric with respect to a finite generating set S.
(2) G is finitely generated and hyperbolic relative to P, S a finite relative generating set, and d the word metric with respect to S ∪ P.
(3) G is the mapping class group Mod(S) and d a metric that is equivariantly quasi-isometric to the curve complex CC(S).
(4) G is Out(F n ) and d a metric that is equivariantly quasi-isometric to the free factor complex F n .

Then (respectively)
(1) if H is quasiconvex, then (G, {H}) has graded relative hyperbolicity; conversely, if (G, {H}) has geometric graded relative hyperbolicity then H is quasiconvex, (2) 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, (3) 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), (4) 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 ).
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 [DGO17]. We will need to generalize the notion of hyperbolic embeddedness in [DGO17] 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: (1) Intersections of conjugates of subgroups in a proper (Γ, d) (the Cayley graph of the ambient group with respect to a finite generating set).
(2) Intersections of metric thickenings of cosets in a not necessarily proper (Γ, d).
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 4.3): (1) Graded relative hyperbolicity (algebraic).
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 4.5, Theorem 4.6, Theorem 4.10, Theorem 4.14 and Proposition 4.15. This, we hope, will clarify our intent.
Definition 2.1 ( [Far98]). -Let (X, d) be a geodesic length space, and be a collection of subsets of X. The electrification (X el Y , d el Y ) of (X, d) along Y is defined as the following coned-off space: The metric d el Y is defined as the path metric on X el Y for the natural quotient metric coming from the product metric on Y i × [0, 1] (defined as above).
Let Y i ∈ Y . The angular metric d Yi (or simply, d, when there is no scope for confusion) on Y i is defined as follows: For y 1 , y 2 ∈ Y i , d Yi (y 1 , y 2 ) is the infimum of lengths of paths in X el Y 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 .
Definition 2.2. -Consider a geodesic metric space (X, d) and a family of subsets Y = {Y i , i ∈ I}. We will say that Y is coarsely hyperbolically embedded in (X, d), if there is a function ψ : R + → R + which is proper (i.e., lim +∞ ψ(x) = +∞), and such that (1) the electrified space X el Y is hyperbolic, (2) the angular metric at each Y ∈ Y in the cone-off is bounded from below by ψ • d| Y .
Remark 2.3. -This notion originates from Osin's [Osi06a], and was developed further in [DGO17] in the context of groups, where one requires that each subset Y i ∈ Y 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 Y 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 [DGO17] only if it is finite.
As in the point of view of [Osi06a], we say that (X, d) is strongly hyperbolic relative to the collection Y (in the sense of spaces) if Y 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. Druţu 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.

Quasiretractions
For the next proposition, define the D-coarse path metric on a subset Y of a pathmetric 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 [DGO17], with a similar proof.
Proposition 2.5. -Let (X, d) be a graph. Assume that Y is coarsely hyperbolically embedded in (X, d). Then, there exists D 0 such that for all Y ∈ Y , the D 0 -coarse path metric on Y ∈ Y is undistorted (or equivalently the D 0 -coarse path metric on Y ∈ Y is quasi-isometric to the metric induced from (X, d)).
We will need the following lemma, which originates in Lemma 4.29 in [DGO17]. The proof is the same; for convenience we will briefly recall it. The lemma provides quasiretractions onto hyperbolically embedded subsets in a hyperbolic space.
Lemma 2.6. -Let (X, d) be a geodesic metric space. There exists C > 0 such that whenever Y is coarsely hyperbolically embedded (in the sense of spaces) in (X, d), then for each Y ∈ Y , there exists a map r : X → Y which is the identity on Y and such that d(r(x), r(y)) Cd(x, y).
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 d(r(x), r(y)) C. The constant C will be 10(δ + 1) + 1. Assume that x and y are at distance > 5(δ + 1) from the cone point. By hyperbolicity, one can find two points in the triangle (p, x, y) at distance 2(δ + 1) from r(x), r(y) at distance 2δ from each other. This provides a path of length at most 6δ + 4. Hence d(r(x), r(y)) 6δ + 4. If x and y are at distance 5(δ + 1) from the cone point, then d(r(x), r(y)) d(r(x), x) + d(x, y) + d(y, r(y)) 2 × 5(δ + 1) + 1.
We can now prove the proposition. Proof. -Choose D 0 = ψ(C): for all y 0 , y 1 ∈ Y at distance 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 , . . . , z n , and project that path by r. One gets r(z 0 ), . . . , r(z n ) in Y , two consecutive ones being at distance at most D 0 . This proves the claim.
2.3. 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 [GM08] for a graph).
We consider inductively on Then H (Y ) is defined as the inductive limit of the H k (Y )'s and is called the horoball over Y . Let (X, d) be a graph, and Y be a collection of subgraphs (with the induced metric on each of them). The horoballification of (X, d) over Y is defined to be the space Y is defined as the path metric on X h Y . One can electrify a horoballification X h Y of a space (X, d): one gets a space quasiisometric to the electrification X el Y of X. We record this observation in the following.
Proposition 2.8. -Let X be a graph, and Y be a family of subgraphs. Let X el Y and X h Y be the electrification, and the horoballification as above.
which is the identity on X and sends the cone point of Y i to the cone point of H (Y i ).
Consider the map e : Then e is a quasi-isometry that induces an isometry on X el Proof. -First, note that a geodesic in (X h ) el H (Y ) between two points of X never contains an edge with a vertex of depth 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 el is at distance at most 2 from a point in X, hence also from a point of the image of X el Y .
2.4. 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 ∈ Q, and any geodesic [x, y] in X, any point of [x, y] is at distance at most C from a point of Q.
The aim of this subsection is to establish criteria for hyperbolicity of certain spaces (electrification, horoballification), and related statements on persistence of quasiconvexity 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.

2.5.
Persistence of hyperbolicity and quasiconvexity 2.5.1. The statements. -Here we state the results for which we give arguments in the following two subsubsections.
Proposition 2.10. -Let (X, d) be a hyperbolic geodesic space, C > 0, and Y be a family of C-quasiconvex subspaces. Then X el Y is hyperbolic. If moreover the elements of Y are mutually cobounded, then X h Y is hyperbolic.
In the same spirit, we also record the following statement on persistence of quasiconvexity.
Proposition 2.11. -Given δ, C there exists C such that if (X, d X ) is a δ-hyperbolic metric space with a collection Y of C-quasiconvex, sets. then the following holds: 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 +λ the set of points at distance at most λ from H. We will say that H (∆, ε)-meets Y if there are two points x 1 , x 2 at distance ∆ from each other, and at distance ε∆ from Y and H, and if for all pair of points at distance 20δ from {x 1 , x 2 }, either H or Y is at distance at least ε∆ − 2δ 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 0 such that the c-neighborhood N c (H) is path connected.
Proposition 2.12. -Let (X, d) be hyperbolic, and let Y 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 el Y . Assume also that there exists ε ∈ (0, 1), and ∆ 0 such that for all ∆ > ∆ 0 , wherever H (∆, ε)-meets an item Y in Y , there is a path in H +ε∆ 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), Y , ∆ 0 , ε, the coarse path connection constant, and the quasi-geodesic constant of the last assumption.
Let (X, d) be a metric space, and Y a collection of subspaces. If γ is a path in (X, d), or even in (X el Y , d el Y ), one can define an elementary electrification of γ in (X el Y , d el Y ) as follows: For x 1 , x 2 in γ, both belonging to some Y i ∈ Y , and at distance > 1, replace the arc of γ between them by a pair of edges ( A complete electrification of γ is a path obtained after a sequence of elementary electrifications of subarcs, admitting no further elementary electrifications.
One can de-electrify certain paths. Given a path γ in (X el , is a path in X such that (1) γ is a complete electrification of σ, (2) (σ γ) ∩ Y i is either empty or a (λ, µ)-quasigeodesic in Y i .
Observe that, given a path σ in X el , 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 Yi on the path σ. It might also happen that there is no way of de-electrifying it, if the spaces in Y are not quasiconvex.
We say that a path γ in (X, d) is an electroambient geodesic if it is a deelectrification of a geodesic.
Proof. -The first part is fairly well-known. In some other guise it appears in [Bow12,Prop. 7.4] [Szc98, Prop. 1] [Mj10]. 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 [Mj10]). Indeed, by replacing (or augmenting) any Y ∈ Y by Y × [0, D] glued along Y × {0}, and replacing Y by the family {Y × {D}, Y ∈ Y }, we achieve a D-separated quasiconvex family.
Next, we discuss Proposition 2.11.
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.
The rest of this (subsub)section is devoted to discussing Proposition 2.12. Towards doing this, we will obtain an argument for showing a variant of the second point of Proposition 2.10, namely that in a hyperbolic space (X, d), a family Y of uniformly quasi convex subspaces that is mutually cobounded defines a strong relative hyperbolic structure on (X, d). The second point of 2.10 as it is stated will be however proved in the next subsection.
Lemma 2.13. -Suppose (X, d) is δ-hyperbolic. Let H be a collection of C-quasiconvex D-mutually cobounded subsets. Then for all P 1, there exists ε 0 = ε 0 (C, P, D, δ) such that the following holds: Let β be an electric (P, P )-quasigeodesic without backtracking (i.e., β does not return to any H 1 ∈ H after leaving it) and γ a geodesic in (X, d), both joining x, y. Then, given ε ε 0 there exists D = D(P, ε) such that (1) Similar Intersection Patterns 1: if precisely one of {β, γ} meets an ε-neighborhood N ε (H 1 ) of an electrified quasiconvex set H 1 ∈ H , then the length (measured in the intrinsic path-metric on N ε (H 1 ) ) from the entry point to the exit point is at most D, (2) Similar Intersection Patterns 2: if both {β, γ} meet some N ε (H 1 ) then the length (measured in the intrinsic path-metric on N ε (H 1 ) ) from the entry point of β to that of γ is at most D; similarly for exit points.
Note that Lemma 2.13 above is quite general and does not require X to be proper. The two properties occurring in Lemma 2.13 were introduced by Farb [Far98] in the context of a group G and a collection H of cosets of a subgroup H. The two together are termed 'bounded coset penetration' in [Far98].
Remark 2.14. -In [Mj10], the extra hypothesis of separatedness was used. However, this is superfluous by the same remark on augmentations of elements of Y that we made in the beginning of the proof of Proposition 2.10. Lemma 2.13 may be stated equivalently as the following (compare with 2.23 below) .
If X is a hyperbolic metric space and H a collection of uniformly quasiconvex mutually cobounded subsets, then X is strongly hyperbolic relative to the collection H .
We give a slightly modified version of [Mj10,Lem. 3.15] below by using the equivalent hypothesis of strong relative hyperbolicity (i.e., Lemma 2.13).
The proof of Lemma 2.15 goes through mutatis mutandis for strongly relatively hyperbolic spaces as well, i.e., hyperbolicity of X may be replaced by relative hyperbolicity in Lemma 2.15 above. We state this explicitly below: Corollary 2.16. -Let (X, d) be strongly relatively hyperbolic relative to a collection Y of path connected subsets. Then, for all λ, µ > 0, there exists λ , µ such that any electroambient (λ, µ)-quasi-geodesic is a (λ , µ )-quasi-geodesic in (X, d).
We include a brief sketch of the proof-idea following [Mj10]. Let γ be an electroambient quasigeodesic. By lemma, its electrification γ is a quasi-geodesic in X el Y . Let σ be the electric geodesic joining the end-points of γ. Hence σ and γ have similar intersection patters with the sets Y i [Far98], i.e., they enter and leave any Y i at nearby points. It then suffices to show that an electroambient representative of σ is in fact a quasigeodesic in X. A proof of this is last statement is given in [McM01,Th. 8.1] in the context of horoballs in hyperbolic space (see also Lemmas 4.8, 4.9 and their proofs in [Far98]). The same proof works after horoballification for an arbitrary relatively hyperbolic space.
The proof of Proposition 2.12 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 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.
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 Y 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 2.10 that (X el . Denote by P the set of cone points corresponding to elements of Y and let γ be a (C, C)-quasi-geodesic without backtracking in (X, d el ) with vertices in H ∪P joining x, y ∈ H. By assumption, the collection Y is uniformly C-quasiconvex. Further, by assumption, there exists ε ∈ (0, 1), and ∆ 0 such that for all ∆ > ∆ 0 , wherever H (∆, ε)-meets an item Y in Y , there is a path in H +ε∆ between the meeting points in H that is uniformly a quasigeodesic in the metric (X, d). Hence, for some uniform constants λ, µ, we may (coarsely) (λ, µ)-de-electrify γ to obtain a (λ, µ)electroambient quasigeodesic γ in (X, d), that lies close to H. [Note that the meeting points of H with elements of Y are only coarsely defined. So we are actually replacing pieces of γ by quasigeodesics in H +ε∆ rather than in H itself.] Note that by assumption Y are uniformly quasiconvex in (X, d); and further that they are uniformly mutually cobounded. Hence the space X is actually strongly hyperbolic relative to Y . By Corollary 2.16, it follows that γ is a quasi-geodesic in (X, d), for a uniform constant.
Since this was done for arbitrary x, y ∈ H i, , we obtain that H is D-quasiconvex in (X, d). This finishes the proof of Proposition 2.12 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 +ε i 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 +ε i 's in X satisfies the property that an intersection of any m of them is bounded. We are then through by the induction hypothesis.
2.5.3. 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 2.5.1. 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 [Cou14,Prop. 5.28]), which uses asymptotic cones in order to show hyperbolicity or quasiconvexity of constructions.
We fix a non-principal ultrafilter ω and will use the construction of asymptotic cones with respect to this ultrafilter ω. A few observations are in order here.
In all the following, (X N , x N ) is a sequence of pointed δ N -hyperbolic spaces, with δ N converging to 0. Recall then that the asymptotic cone lim ω (X N , x N ) is an R-tree, with a base point.
If Y 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 ω ((X N ) el Y N , x N ) and relate it to lim ω (X N , x N ). Let us define the following equivalence relation on the set of sequences in X N .
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 Let us call C this set of equivalence classes. For a sequence u = (u N ) we write ∼ u for its class in C. We also allow ourselves to write lim ω (X N , ∼ u ) for lim ω (X N , u N ) to avoid cluttered notation.
for each class ∼ u in C. Given two sequences (y N ), and (z N ) in the same class for ω, 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 C, then d X N (z N , y N ) is not o(1) for ω, and again, in the electric metric, it is not o(1). The map of the lemma is thus injective.
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 Y . One says that the sequence is visible We define Y ω to be the collection of all sets lim ω (Y N , * ), for all possible sequences . This is well defined, and injective, for if c N is another sequence of cone-points ω-almost everywhere different from c N , then it defines an ω-almost everywhere different sequence Y N , and a different set lim ω (Y N , * ). We also can extend our map to all lim ω ((X N ) el Y N , x N ) linearly on the cone-edges. This produces a bijection lim ω (( To show that it is an isometry, consider two sequences (y N ), (z N ) both in X N , such that the distance in X el Y N converges (for ω) to . Then there is a path of length N (converging to ) in X el Y N with, eventually, at most /2 cone points on it. It follows from the construction that their images in ∼u∈C lim ω (X N , ∼ u ) el are at distance at most . Conversely, assume (y N ) and (z N ) are sequences in X N giving points in lim ω (X N , ∼ u ) and lim ω (X N , ∼ v ), for ∼ u , ∼ v ∈ C, and take a path γ between these points in lim ω (( It has finite length, so it contains finitely many cone points of length , by using the corresponding cone points and the path between the spaces Y (i) N given by the restriction of γ. We have thus observed that the bijection we started with is 1-Lipschitz as is its inverse. It is therefore an isometry.
We finally describe a tree-like structure on Let us say that two classes ∼ u and ∼ 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.
Proof. -Note that by Lemma 2.17 the intersection consists of cone points. Thus getting o(1)-close to each other, on sequences that are visible for ∼ u and sequences that are visible for ∼ v . But these sides are close to Y N and Y N respectively. It follows that in lim ω (X N , ∼ u ) and in lim ω (X N , ∼ v ), the limit of Y N and of Y N share a point. Thus the cone point of their electrifications are at distance 2.
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.
As a corollary, Proof. -The number k is fixed, and the argument will generalize the one of the previous lemma. For each i, let (y , using the approximation by a finite tree (for hyperbolic spaces), we see that one of the segments [y After extracting a subsequence, one can assume that i is constant in N , and we choose it to be our i 0 . It follows that the sequence (Y (i0) N ) is visible for ∼ u1 and the limit of (Y (i0) The conclusion that the cone points of (Y el follows, and this also implies that they are at distance at most 2 in Note that if the subsets of Y N are c n -mutually cobounded, with c n going to 0 (or even bounded) then one can improve the lemma by saying that eventually Y From the previous lemmas we get: el is the union of spaces of the form el in γ i such that the distance between c 1 and c 2 is at most 2.
Indeed, such a pair of paths provides us with a certain finite cycle of classes, starting with ∼ u , and we may apply the previous lemma.
We say that the space Let us prove Proposition 2.10.
Proof. -We claim that, for all ρ, there exists δ 0 < ρ/10 14 and C 0 < ρ/10 14 such that if X is δ 0 -hyperbolic, and if Y is a collection of C 0 -quasiconvex subsets, then every ball of radius ρ of X el Y is 10-hyperbolic. For proving the claim, assume it false, and consider a sequence of counterexamples (X N , Y N ) for δ 0 = C 0 = 1/N , N = 1, 2, . . . . This means that (X N ) el Y N fails to be 10-hyperbolic. There are four points We pass to the ultralimit for ω. In lim ω (X el N , x N ), each sequence x N , y N , z N , t N converges, since these points stay at bounded distance from x N , and the inequality persists. Hence one gets four points falsifying the 10-hyperbolicity condition in a pointed space is, by Corollary 2.21, a 2-quasi-tree of spaces that are electrifications (of parameter 1) of real trees lim ω (X N , x N ), for some base point x N over the family Y ω , consisting of convex subsets (i.e., of subtrees). This We now claim that, under the same hypothesis, it is (2 + 10C 0 + 10δ 0 )-coarsely simply-connected, that is to say that any loop in it can be homotoped to a point by a sequence of substitutions of arcs of length < (2 + 10C 0 + 10δ 0 ) by its complement in a loop of length < (2 + 10C 0 + 10δ 0 ). Indeed, any time such a loop passes through a cone point associated to some Y ∈ Y , one can consider a geodesic in X between its entering and exiting points in Y , which stays in the C 0 neighborhood of Y . Therefore, a (2 + 10C 0 )-coarse homotopy of the loop transforms it into a loop in X, which is δ-hyperbolic. Since a δ-hyperbolic space is 10δ-coarsely simply-connected, the second claim follows.
The final ingredient is the Gromov-Cartan-Hadamard theorem [Cou14, Th. A.1], stating that, if ρ is sufficiently large compared to µ, any ρ-locally 10-hyperbolic space which is µ-coarsely simply connected is (globally) δ -hyperbolic, for some δ . We thus get that there exists δ 0 < ρ/10 14 and C 0 < ρ/10 14 such that if X is δ 0 -hyperbolic, and if Y is a collection of C 0 -quasiconvex subsets, then X el Y is δ -hyperbolic. Now let us argue that this implies the first point of the proposition. If X and Y are given as in the statement, one may rescale X by a certain factor λ > 1, so that it is δ 0 -hyperbolic, and such that Y is a collection of C 0 -quasiconvex subsets. Let us is endowed with the product metric as defined in the first paragraph of 2.1 except that 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 ρ, there exist δ 0 , C 0 and D 0 such that if X is δ 0 -hyperbolic, and if Y is a collection of C 0 -quasiconvex subsets, D 0 -mutually cobounded, then any ball of radius ρ of the horoballification X h Y is 10-hyperbolic. The proof of the claim is similar. Consider a sequence of counterexamples X N , Y N , for the parameters δ = C = D = 1/N for N going to infinity, with the four points in a ball of radius ρ, falsifying the hyperbolicity condition. There are two cases. Either x N (which is in (X N ) h Y ) escapes from X N , i.e., its distance from some basepoint in X N tends to ∞ for the ultrafilter ω, or it does not. In the case that it escapes from X N , then, when it is larger than ρ 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 ∈ X N whose distance to x N remains bounded (for the ultrafilter ω).
is the horoballification of the asymptotic cone of the sequence (X N , x N ) over the family lim ω (Y N , x N ) defined above.
This family lim ω (Y N , x N ) consists of 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 [DS05], with pieces being the combinatorial horoballs over the subtrees constituting lim ω Y N . As a tree of 10-hyperbolic spaces, this space is 10-hyperbolic, contradicting the inequalities satisfied by the limits lim ω {x N , y N , z N , t N }. Therefore, X h Y is ρ-locally 10-hyperbolic. As before, one may check that (under the same assumptions) X h Y is (2 + 10C 0 + 10δ 0 )-coarsely simply connected, and again this implies by the Gromov-Cartan-Hadamard theorem that (under the same assumptions) X h Y is hyperbolic.
This implies the second point. Indeed, let us denote by 1 λ X the space X with metric rescaled by 1/λ.
The previous claim shows that, under the assumption of the second point of the proposition, there exists λ > 1 such that λ Y that is identity on X and that sends {y} × {n} to {y} × {λ × (n + log 2 λ )} for all y ∈ Y i and all Y i (and all n). All paths in X h Y that have only vertical segments in horoballs have their length expanded (under the map η) by a factor between 1 and λ + log 2 λ. But the geodesics in X h Y and λ( 1 are paths whose components in horoballs consist of a vertical (descending) segment, followed by a single edge, followed by a vertical ascending segment (see [GM08]). Hence η is a quasi-isometry, and the space X h Y is hyperbolic.
We continue with the persistence of quasi-convexity of Proposition 2.11.
Proof. -The strategy is similar to that in the previous proposition. The main claim is that for all ρ, there is δ 0 < 1, then Q is ρ-locally 10-quasiconvex in X el Y (of course δ 0 , C 0 will be very small). To prove the claim, again, by contradiction, consider a sequence X N , Y N , Q N of counter examples for δ N = C N = 1/N for N = 1, 2, . . . . There exist two points x N , y N in Q N , at distance ρ from each other (for the electric metric), and a geodesic [x N , y N ] in X el Y with a point z N on it at distance > 10 from Q. We record a point z N in Q at minimal distance ( ρ in any case) from z N .
With a non principal ultrafilter ω, we may take the asymptotic cone of the family of pointed spaces (X el N , x N ). In lim ω ((X N ) el Y , 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 ω (Q N , x N ) is not ρ-locally 10-quasiconvex in lim ω ((X N ) el Y , x N ). But as we noticed in Corollary 2.21 lim ω ((X N ) el Y , x N ) is a 2-quasi-tree of spaces of the form (lim ω (X N , x N )) el Y ω , which are the electrifications of R-trees lim ω (X N , x N ) over a family of convex subsets (i.e., subtrees). In this space, lim ω (Q N , x N ) is also a subforest of (u N )∈C lim ω (X N , u N ). Also observe that if (Q N ) is visible in two adjacent classes, then lim ω (Q N , x N ) is adjacent to their common cone point over sequences , and this contradicts the inequalities satisfied by lim ω {x N , y N , z N , z N }. The claim is established for all ρ.
Now there exists ρ 0 such that, in any 1-hyperbolic space, any subset that is ρ 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 ρ, we have proven that there is δ 0 < 1, C 0 < 1 and C 1 , such that if (X, d X ) is δ 0 -hyperbolic, if 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 el Y . Coming back to the statement of the proposition, by rescaling our space, we have proven that if (X, d X ) is a δ-hyperbolic metric space with a collection Y of C-quasiconvex sets, and if Q is C-quasiconvex, then Q is λC 1 -quasiconvex in X el λ Y (as defined in the previous proof) for λ = max{δ for C depending only on δ, C.
Finally, we consider the proposed converse in Proposition 2.12. 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 ⊂ 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 , . . . , x r = y for some r such that x i ∈ Y and d X (x i , x i+1 ) m for all i. We call such a sequence an m-coarse path or a path with mesh m. The length of the coarse path (x 0 , . . . , 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.
Then there exists δ 0 , C 0 , m 0 > 0, such that the following holds: Assume that (X, d) is geodesic, δ 0 -hyperbolic, with a collection Y of C 0 -quasiconvex subsets. Further suppose that H is an m 0 -coarsely connected subset of X which is C el H -quasiconvex in the electrification X el Y . Equip H +ε∆ with its m 0 -coarse path metric d H .
Assume also that whenever H (∆, ε)-meets a set Y ∈ Y , there is a (Q, C 0 )-quasigeodesic path, which is an (m 0 /10)-coarse path, in H +ε∆ joining the meeting points in H.
Then for all a, b ∈ H +ε∆ at d H -distance at most R from each other, any m 0 -coarse δ 0 -quasi-geodesic of H +ε∆ (for its coarse path metric) between a, b is (∆ × C el H )-close to a geodesic of X.
Proof. -Suppose that the claim is false: For all choice of δ, C, m there is a counterexample. Set δ N = C N = 1/N . For each ε, there exists N such that, in a (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 ε-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 (9∆/10)-long (Q, 1/N )-quasigeodesics with mesh 1/N in a (1/N )-hyperbolic spaces, with starting points at distance ∆/10 from each other, and ending points at distance ∆/10 from each other, necessarily lie at distance (m N /10) from one another.
Let then X N , H N , Y N be a counterexample to our claim for these values: Passing to an asymptotic cone, we find a map p ω from an interval [0, R] to a continuous path in lim ω (H N , a N ) from a ω to b ω (which can be equal to a ω ) that passes through a point c ω at distance (∆ × C el H ) from the arc [a ω , b ω ] in lim ω (X N , a N ). However, it is at distance C el H in the electrification of lim ω (X N , a N ) by Y ω . It follows that on the path in lim ω (X N , a N ) from [a ω , b ω ] to c ω , there must exist a segment of length ∆ belonging to the same Y ω ∈ Y ω . Let us say that Y ω is the limit of a sequence Y N . Note that the limit path p ω crosses this segment at least twice (once in either direction).
Thus, for N large enough, H N (∆, ε)-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 ) 9∆/10 (and (s 1 , s 2 ) 9∆/10) and d(r 1 , s 1 ) 3∆/10 and d(r 2 , s 2 ) 3∆/10. By assumption, there is a (Q, 1/N )-quasigeodesic path in H +ε∆ from s 1 to s 2 and another from r 1 to r 2 , with mesh < m N /10. They have to fellow travel on a large subpath, and pass at distance m N /10 from each other, by choice of m N . One can therefore find a shortcut that is still a path in H +∆ of mesh m N , a contradiction.
From the claim, we can prove the statement of the proposition. Consider a situation as in the statement. We may choose the coarse path connectivity constant of H to be more than 10 times the quasigeodesic constant of the last assumption there. Take ε, given by the assumption of the proposition, and ∆ > max{100ε, ∆ 0 }. Let Q be the quasi-geodesic constant given by the the assumption of the proposition on (∆, ε)-meetings, and C el H 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 Y , and the constant of coarse path connection of H are respectively smaller than δ 0 , C 0 , m 0 of the lemma above.
Note that the assumption of the proposition on (∆, ε)-meetings is invariant under rescaling (except for the value of ∆ 0 ). Thus, this assumption still holds, with the same ε, and for the specified ∆ chosen above. The lemma applies, and H +ε∆ is R-locally quasiconvex for the rescaled metric. By the local to global principle (in δ 0 hyperbolic spaces), with a suitable preliminary (large enough) choice of R, H +ε∆ is then globally quasiconvex. After rescaling back to the original metric of X, H +λ is still quasiconvex for some λ (depending on ε∆, and the coefficient of rescaling); hence H is quasiconvex.
By construction, we also have the statement on the dependence of the quasiconvexity constant.
2.5.4. 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.
Proposition 2.23. -Assume that (X, d) is a metric space, and that Y is a collection of subspaces.
If the horoballification X h Y of X over Y is hyperbolic, then Y is coarsely hyperbolically embedded in the sense of spaces.
If X is hyperbolic and if Y is coarsely hyperbolically embedded in the sense of spaces, then X h Y is hyperbolic.
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.
Proof. -Assume that the horoballification X h Y of X over Y is δ-hyperbolic. The horoballs Y h (corresponding to Y ) are thus 10δ-quasiconvex. Therefore, by Proposition 2.10, the electrified space obtained by electrifying (coning off) the horoballs Y h is hyperbolic.
Since by Proposition 2.8 this space is quasi-isometric to (X el Y , d el Y ), it follows that the later is hyperbolic. This proves the first condition of Definition 2.2.
We want to prove the existence of a proper increasing function ψ : R + → R + , such that the angular metric at each cone point Of course, the angular metric at v Y is bounded below by ψ • d| Y . The function ψ is obviously increasing. We need to show that it is proper, i.e., that it goes to +∞.
If ψ is not proper, then there exists θ 0 > 0 such that for all D, there exist Y ∈ Y and y, y ∈ Y at d-distance greater than D but d(y, y ) θ 0 (where d is the angular metric on Y ). We choose D θ 0 δ (for instance D = exp(100(θ 0 + 1)(δ + 1))). Consider a path in (X h Y ) el of length less than θ 0 from y to y avoiding the cone point of Y . Because D θ 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 , . . . , v N (corresponding to Y 1 , . . . , Y N with N < θ 0 ). Adjoining where v Y corresponds to the cone point for Y ), we thus have a geodesic (N + 2)-gon σ. Next replace each passage of σ through a cone point The geodesic segments µ i or µ Y comprise (n + 1) alternate sides of this geodesic (2N + 2)-gon.
Since X h Y is δ-hyperbolic, it follows that the mid-point m of µ Y is at distance (2N + 2)δ 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 θ 0 and δ: D exp(4(N + 1)δ). Since N θ 0 , this is a contradiction with the choice of D. We can conclude that ψ is proper, and we have the first statement. Let us consider the second statement. If X is hyperbolic and if Y is coarsely hyperbolically embedded in the sense of spaces, then elements of Y are uniformly quasiconvex in (X, d) by 2.7, and, by the property of the angular distance on any Y ∈ Y , they are mutually cobounded. The statement then follows by Proposition 2.10.

Algebraic height and intersection properties
3.1. Algebraic height. -We recall here the general definition for height of finitely many subgroups.
Definition 3.1. -Let G be a group and {H 1 , . . . , H m } be a finite collection of subgroups. Then the 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 1 H α1 , . . . , g n+1 H αn+1 , the intersection 1 i n+1 g i H αi g −1 i is finite.
We shall describe this briefly by saying that algebraic height is the largest n for which the intersection of n essentially distinct conjugates of H 1 , . . . , H m is infinite. Here 'essentially distinct' refers to the cosets of H 1 , . . . , H m and not to the conjugates themselves.
For hyperbolic groups, one of the main theorems of [GMRS98] is the following: We quickly recall a proof of Theorem 3.2 for one subgroup H in order to generalize it to the context of mapping class groups and Out(F n ).
Proof. -Let G be hyperbolic, X(= Γ G ) a Cayley graph of G with respect to a finite generating set (assumed to be δ-hyperbolic), and H a C 0 -quasiconvex subgroup of G. Suppose that there exist N essentially distinct conjugates {H gi }, i = 1, . . . , N , of H that intersect in an infinite subgroup. The N left-cosets g i H are disjoint and share an accumulation point p in the boundary of G (in the limit set of i H gi ). Since all g i H are C 0 -quasiconvex, there exist N disjoint quasi-geodesics σ 1 , . . . , σ N (with same constants λ, µ depending only on C 0 , δ) converging to p such that σ i is in g i H. Since X is δ-hyperbolic, there exists R(= R(λ, µ, δ) = R(C 0 , δ)) and a point p 0 sufficiently far along σ 1 such that all the quasi-geodesics σ 1 , . . . , σ N pass through B R (p 0 ). Hence N #(B R (p 0 )) giving us finiteness of height.
Further, any such σ i furnishes a double coset representative g i of g i (say by taking a word that gives the shortest distance between H and the coset g i H) of length bounded in terms of R. This furnishes the second conclusion.
Remark 3.3. -A word about generalizing the above argument to a family H of finitely many subgroups is necessary. The place in the above argument where H consists of a singleton is used essentially is in declaring that the N left-cosets g i H 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 gi i }, i = 1, . . . , N , H i ∈ H must contain N essentially distinct conjugates {H gi }, i = 1, . . . , N of some H ∈ H and then the above argument for a single H ∈ H goes through.
Remark 3.4. -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.  For i ∈ N, i 2, define the geometric i-fold intersection, or simply the i-fold intersection of cosets of H , H i , to be the set of subsets J of G for which there exist H 1 , . . . , H i ∈ H and g 1 , . . . , g i ∈ G, and ∆ ∈ N satisfying:

Geometric
+∆ is not in the 20δ-neighborhood of (g j H j ) +∆−2δ , and the diameter of J is at least 10∆.
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.
Proposition 3.6. -Consider J ∈ H j , and K ∈ H k for j < k and let ∆ J , ∆ K be constants as in Definition 3.5 for defining J and K respectively.
Assume that J and K (∆, ε)-meet, for some ∆ > 20∆ 0 , and for ε < 1/50. Then either K ⊂ J, or for any pair of (∆, ε)-meeting points of J and K, there is L ∈ H j+1 contained in J, that contains it. Proof.
-Let x, y be (∆, ε)-meeting points of J and K. If K ⊂ J, we can assume that x, y are in (g 1 H 1 ) +∆ K +ε∆ for some g 1 H 1 not contained in the collection {g i H i }.
We argue by contradiction. Suppose that x, y are contained in the 20δ-thickening of a 2δ-lesser intersection. It follows that there are x , y such that d(x, x ) 20δ, d(y, y ) < 20δ and still d(x , J) ε∆ − 2δ, and d(y , J) ε∆ − 2δ. But by definition of (∆, ε)-meeting, this is a contradiction.
Finally, the diameter of the intersection of ∆ -thickenings of our cosets, is larger than ∆. Since the thickening constant is ∆ ∆ 0 + ε∆, the ratio of the thickening constant by the diameter is at most (∆ 0 + ε∆)/∆ which is less than 1/10, hence the result.
Let (G, d) be a group with a word metric and H < G a subgroup. The restriction of d on H will be called the induced metric on H from G.
Proposition 3.7. -Let (G, d) be a group with a δ-hyperbolic word metric (not necessarily locally finite).
Assume that A 1 , . . . , A n are C-quasiconvex subsets of G. Then for all ∆ > C +20δ,

Moreover, if A and B are C-quasiconvex subsets of G, and if Π B (A) denotes the set of nearest points projections of
Proof. -Consider x, y ∈ A +∆ i and take a i , b i some nearest point projection on A i . On a geodesic [x, y] take p at distance greater than 4δ from x and y. Hyperbolicity applied to the quadrilateral (x, a i , b i , y) tells us that -If p is close to an approaching segment of index i, then it is in A +∆ i . -If x is close to the traveling segment of index i, then it is at distance at most We thus obtain that [x, y] remains at distance 4δ from A +∆ i . To prove the second statement, take a 0 , b 0 in A and B respectively realizing the distance (up to δ if necessary). Let b ∈ Π 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 3.3. 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. [GMRS98].
Given a finite family of subgroups of a group we first define collections of i-fold conjugates or algebraic i-fold intersections.  H 1 , . . . , H i ∈ H and g 1 , . . . , g i ∈ G satisfying: -the cosets g j H j are pairwise distinct (and hence as in [GMRS98] we use the terminology that the conjugates {g j H j g −1 j , j = 1, . . . , i} are essentially distinct) We shall call H i the family of algebraic i-fold intersections or simply, i-fold conjugates.
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-translate of the nearest point projection of cH 2 onto H 1 . Let Π B (A) denote the (nearest-point) projection of A onto B. Then Π H1 (cH 2 ) lies in a bounded neighborhood (say D 0 -neighborhood) of H c 2 ∩ H 1 and so Π aH1 (bH 2 ) lies in a D 0 -neighborhood of bH 2 b −1 a ∩ aH 1 . The latter does lie in a bounded neighborhood of (H 2 ) b ∩ (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 [Mj10,Mj14] while the notion of geometric i-fold intersections discussed earlier is in the spirit of [DGO17]. (1) for all n, and all H ∈ H n , H has a conjugate H such that if d is any induced metric on H from some H i ∈ H , then (H , d ) is (C 1 , C 1 )-qi-embedded in (G, d), (2) for all n, let (H n ) 0 be a choice of conjugacy representatives of elements of H n that are C 1 -quasiconvex in (G, d); Let C H n denote the collection of left cosets of elements of (H n ) 0 ; for all A, B ∈ C H n with A = aA 0 , B = bB 0 , and A 0 , B 0 ∈ (H n ) 0 , Π B (A) either has diameter bounded by C n for the metric d, In keeping with the spirit of the previous subsection, we provide a geometric version of the above definition below.  H n , (H, d) is C n -coarsely path connected, and (C 1 , C 1 )qi-embedded in (G, d) (for its coarse path metric), (2) for all A, B ∈ H n either diam G,d (Π B (A)) C n , or Π B (A) ⊂ A +Cn for d.
Remark 3.11. -The second condition of Definition 3.10 follows from the first condition if d is hyperbolic by Proposition 3.7. Further, the first condition holds for such (G, d) so long as ∆ is taken of the order of the quasiconvexity constants (again by Proposition 3.7). 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 3.6 the collection of geometric n-fold intersections H n is mutually cobounded for the metric of (G, d) el

3.4.
Existing results on algebraic intersection properties. -We start with the following result due to Short.
We remark here that in the original statement of [Sho91,Prop. 3], 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 3.12.
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 4.3 and 4.4 below for the lemmas) are independent of the finite generating sets chosen. Hence we have the following.
Theorem 3.13. -Let G be either Mod(S) or Out(F n ) equipped with some finite generating set.
The corresponding statement for relatively hyperbolic groups and relatively quasiconvex groups is due to Hruska. For completeness we recall it.
Definition 3.14 ( [Osi06b, Hru10]). -Let G be finitely generated hyperbolic relative to a finite collection P of parabolic subgroups. A subgroup H G is relatively quasiconvex if the following holds.
Let S be some (any) finite relative generating set for (G, P), and let P be the union of all P i ∈ P. Let Γ denote the Cayley graph of G with respect to the generating set S ∪ P and d the word metric on G. Then there is a constant C 0 = C 0 (S, d) such that for each geodesic γ ⊂ Γ joining two points of H, every vertex of γ lies within C 0 of H (measured with respect to d).
Theorem 3.15 ([Hru10]). -Let G be finitely generated hyperbolic relative to P. 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 ∩ H 2 is C 1 -relatively quasiconvex in G.

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.
Definition 4.1. -Let G be a group, with a left invariant word metric d(= d G ) with respect to some (not necessarily finite) generating set. Let H be a family of subgroups of G.
The geometric height, of H in (G, d) (for d) is the minimal number i 0 so that the collection H 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}.
Remark 4.2 (Comparing notions of height) -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.
-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.
-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.
-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.
We generalize Definition 1.3 of graded relative hyperbolicity to the context of geometric height as follows. We say that G is graded geometric relatively hyperbolic with respect to C H N if (1) H has geometric height n for some n ∈ N, and for each i there are finitely many orbits of i-fold intersections, (2) for all i n + 1, C H i−1 is coarsely hyperbolically embedded in (G, d i ), (3) there is D i such that all items of C H i are D i -coarsely path connected in (G, d).
Remark 4.4 (Comparing geometric and algebraic graded relative hyperbolicity) Note that the second condition of Definition 4.3 is equivalent, by Proposition 2.23, to saying that (G, d i ) is strongly hyperbolic relative to the collection H i−1 . This is exactly the third (more algebraic) condition in Definition 1.3. Also, the third condition of Definition 4.3 is the analog of (and follows from) the second (more algebraic) condition in Definition 1.3.
Thus finite geometric height along with (algebraic) graded relative hyperbolicity implies graded geometric relative hyperbolicity.   (G, d), and if N is the cardinality of a ball of (G, d) of radius 2C + 10δ, and if g 0 H, . . . , g k H are distinct cosets of H for which there exists ∆ such that the total intersection k i=0 (g i H) +∆ has diameter more than 10∆, and more that 100δ, then there exists x ∈ G such that each g i H intersects the ball of radius N around x.
First note that the second statement implies the first in the (by the third point of Remark 4.2). 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 + 10δ).
Proof. -Assume that there exists ∆ > 0, and elements 1 = g 0 , g 1 , . . . , g k for which the cosets g i H are distinct, and k i=0 (g i H) +∆ has diameter larger than 10∆ and than 100δ.
First we treat the case ∆ > 5δ. Pick y 1 , y 2 ∈ k i=0 (g i H) +∆ at distance 10∆ from each other, and pick x ∈ [y 1 , y 2 ] at distance larger than ∆ + 10δ from both y i . For each i an application of hyperbolicity and quasi-convexity tells us that x is at distance at most 2C + 10δ from each of g i H. The ball of radius 2C + 10δ around x thus meets each coset g i H.
If ∆ 5δ, we pick y 1 , y 2 ∈ k i=0 (g i H) +∆ at distance 100δ from each other, and take x at distance greater than 10δ from both ends. The end of the proof is the same.

4.2.
Relatively hyperbolic groups. -If G is hyperbolic relative to a collection of subgroups P, then Hruska and Wise defined in [HW09] 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 , . . . , g n such that the g i H are (n + 1)-distinct cosets, the intersections of conjugates i g i Hg −1 i 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 P. 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. P) is relatively quasiconvex. It is also true [DGO17] that the normalizer of any P ∈ P is itself and that the subgroup generated by any P and any infinite order element g ∈ G P contains the free product of conjugates of P by g kn , k ∈ 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. P). 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 , . . . , H k , as in the case for the algebraic height.
Hruska and Wise proved that relatively quasiconvex subgroups have finite relative height. More precisely: Theorem 4.6 ([HW09, Th. 1.4, Cors. 8.5-8.7]). -Let (G, P) be relatively hyperbolic, let S be a finite relative generating set for G and Γ be the Cayley graph of G with respect to S. Then for σ 0, there exists C 0 such that the following holds.
Let H 1 , . . . , H n be a finite collection of σ-relatively quasiconvex subgroups of (G, P). Suppose that there exist distinct cosets {g m H αm } with α m ∈ {1, . . . , n}, m = 1, . . . , n, such that m g m H αm g −1 m is not contained in a parabolic P ∈ P. Then there exists a vertex z ∈ G such that the ball of radius C in Γ intersects every coset g m H αm .
Further, for any i ∈ {1, . . . , n}, there are only finitely many double cosets of the form is not contained in a parabolic P ∈ P. Let G be a relatively hyperbolic group, and let H be a relatively quasiconvex subgroup. Then H has finite relative algebraic height.
This allows us to give an example of geometric height in our setting.
Proposition 4.7. -Let (G, P) be a relatively hyperbolic group, and (G, d) a relative word metric d (i.e., 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 P, and hence, in general, not a finite generating set). Let H be a relatively quasiconvex subgroup. Then, H has finite geometric height for d.
This just a rephrasing of Hruska and Wise's result Theorem 4.6. The proof is similar to that in the hyperbolic groups case, using for instance cones instead of balls.

4.3.
Mapping class groups. -Another source of examples arise from convexcocompact 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 CC(S) will denote respectively the Teichmüller space and Curve Complex of S. One important ingredient in Kent-Leininger's proof of Theorem 4.9 is a lifting of the limit set of H in ∂ CC(S) (the boundary of the curve complex) to ∂ Teich(S) (the boundary of Teichmüller space). What is important here is that Teich(S) is a proper metric space unlike CC(S). Further, they show using a theorem of Masur [Mas80], that any two Teichmüller geodesics converging to a point on the limit set Λ H (in ∂ Teich(S)) of a convex cocompact subgroup H are asymptotic. An alternate proof is given by Hamenstädt in [Ham10]. With these ingredients in place, the proof of Theorem 4.10 below is an exact replica of the proof of Theorem 3.2 above: The collection {H 1 , . . . , H n } has finite algebraic height.
A more geometric strengthening of Theorem 4.10 can be obtained as follows using recent work of Durham and Taylor [DT15], 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 [DT15]. Moreover, any σ-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.
Proof. -Assume that the theorem is false: there exists σ such that for all k, and all D, there exists a σ-convex cocompact subgroup H, with a collection of distinct cosets {g m H, m = 0, . . . , k} (with g 0 = 1), satisfying the property that m (g m H) +∆ has diameter larger than max{10∆, D}.
Let a, b be two points in m (g m H) +∆ such that d(a, b) max{10∆, D}. For each i, let a i , b i in g i H be at distance at most ∆ from a and b respectively. Consider γ i geodesics in H from g −1 i a i to g −1 i b i . Consider also a i and b i -nearest point projections of a 0 and b 0 on g i γ i . Finally, denote by g i γ i the subpath of g i γ i between a i and b i By [DT15,Prop. 5.7], H is quasiconvex in G (for a fixed chosen word metric), and for each i, g i γ i is a f (σ)-quasi-geodesic (for some function f ).
We thus obtain from a 0 to b 0 a family of k + 1 paths, namely γ 0 and (one for all i), the concatenation η i = [a 0 , a i ] · g i γ i · [b i , b 0 ]. For D large enough, the paths η i are 2f (σ)-quasigeodesics in G.
Stability of H ([DT15, Th. 1.1]) implies that, there exists R(σ) such that in G, the paths remain at mutual Hausdorff distance at most R(σ). This is thus also true in the Teichmüller space by the orbit map. Hence it follows that all the subpaths g i γ i are at distance at most 2R(σ) 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. (1) all non-trivial elements of H are atoroidal and fully irreducible, (2) for some (any) x ∈ cv n , the (projectivized) Outer space for F n , the orbit Hx ⊂ cv n is σ-quasiconvex with respect to the Lipschitz metric.
Remark 4.12. -The above lemma, while not explicit in [DT14], is implicit in Section 1.2 of that paper.
Also, a word about the metric on cv n is in order. The statements in [DT14] are made with respect to the 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 [DT14] apply via this quasi-isometry.
The following theorem gives a characterization of convex cocompact subgroups in this context and is the analog of Theorem 4.9.
Theorem 4.13 ( [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 ∈ F n (the free factor complex of F n ), the orbit Hx ⊂ F n is qi-embedded in F n .
Dowdall and Taylor also show [DT14,Th. 4.1] that any two quasi-geodesics in cv n converging to the same point p on the limit set Λ H (in ∂cv n ) of a convex cocompact subgroup H are asymptotic. More precisely, given λ, µ and p ∈ Λ H there exists C 0 (= C 0 (λ, µ, p)) such that any two (λ, µ)-quasi-geodesics in cv n converging to p are asymptotically C 0 -close. As observed before in the context of Theorem 4.10, this is adequate for the proof of Theorem 4.10 to go through: Theorem 4.14. -Let G = Out(F n ), and cv n the Outer space for G with a base point z 0 . Then for σ 0, there exists C 0 such that the following holds.
Let H 1 , . . . , H n be a finite collection of σ-convex cocompact subgroups of G. Suppose that there exist distinct cosets {g m H αm } with α m ∈ {1, . . . , n}, m = 1, . . . , n, such that m g m H αm g −1 m is infinite. Then there exists a point z ∈ cv n such that the ball of radius C in cv n intersects every image of z 0 by a coset g m H αm z 0 .
Further, for any i ∈ {1, . . . , n}, there are only finitely many double cosets of the form The collection {H 1 , . . . , H n } has finite algebraic height.
Since an analog of the stability result of [DT15] in the context of Out(F n ) is missing at the moment, we cannot quite get an analog of Theorem 4.11.

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. (1) 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.
(2) Let H be a relatively quasiconvex subgroup of a relatively hyperbolic group (G, P). Let P 0 be a set of conjugacy representatives of groups in P, and d a word metric on G over a generating set S = S 0 ∪ P 0 , where S 0 is finite. Then {H} satisfies the uniform (geometric) qi-intersection property with respect to d.
(3) Let H be a convex-cocompact subgroup of the Mapping Class Group Mod(Σ) of an oriented closed surface Σ of genus 2. If d is a word metric on Mod(Σ) that makes it quasi-isometric to the curve complex of Σ, then H satisfies the uniform (geometric) qi-intersection property with respect to d.
(4) Let H be a convex-cocompact subgroup of Out(F n ) for some n 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.
Proof. -All four cases have similar proofs. Consider the first point.
Case 1: G hyperbolic, H quasiconvex. -Let h be the height of H (which is finite by Theorem 3.2): every h+1-fold intersection of conjugates of H is finite, but some h-fold intersection is infinite.
-The first conditions of Definition 3.9 and Definition 3.10 follow from this finiteness and Proposition 4.5 and Theorem 3.12.
-The second condition of Definition 3.10 follows from Proposition 3.7.
-We prove the second condition of Definition 3.9 (on mutual coboundedness of elements of C H i ) iteratively.
By Theorem 3.12, there exists C h such that two elements of C H 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 × B, such that a −1 i b i are elements of length at most 20δC 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 −1 i b i take the same value. It follows that there are two essentially distinct conjugates of elements of H 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 C H h are mutually cobounded.
We continue by descending induction. Assume that the second property of Definition 3.9 is established for C H i+1 . By Proposition 2.10 it follows that (G, d i+1 ) is hyperbolic. Let δ i+1 be its hyperbolicity constant. By Proposition 2.11, there exists C i such that two elements of C H i are C i -quasiconvex in (G, d i+1 ).
Again take A and B two distinct elements of C H i such that the projection of A on B has diameter greater than D > 1000δ i+1 for d i+1 . Then there are at least D/C i pairs of elements (a i , b i ) in A × B, such that a −1 i b i is an element of length at most 20δ i+1 C h for the metric d i+1 , and for all i there exists i such that the segments )-far from one another. Apply the Proposition 2.12 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 2.15). 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 (200δC 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 (200δC h ) in the metric d) such that for D/( is larger than the cardinality of the finite order elements of G. It follows that the two essentially distinct conjugates of elements of H 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 H j for j i + 1, but therefore must have diameter 2 in the metric d i+1 .
Case 2: G relatively hyperbolic, H relatively quasiconvex. -The geometric height of H for the relative metric is finite, by Proposition 4.7. Let h be its value. The first points of Definition 3.9 follows from this finiteness and Theorem 3.15.
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 C H h such that the projection of A on B has diameter greater than D, then there are D/C H pairs of elements such that both segments are short (for d) and are at distance at least (50δC 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 (100δC h ) + 2(2C h + 5δ). Indeed, consider α and β quasi-geodesic paths in A and B respectively, from a i to a j and from b i to b j . By hyperbolicity and quasigeodesy, at distance 30δC h from a i and b i , there is a path of length 2(2C h + 5δ) joining α to β. 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 × (30δC h ) + 2(2C h + 5δ) that does not intersect [a i , b i ] outside its end points. It follows indeed that the maximal angle of 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 −1 i 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 3.9 is established for C H 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 2.10 it follows that (G, d i+1 ) is hyperbolic. Let δ i+1 be its hyperbolicity constant. By Proposition 2.11, there exists C i such that two elements of C H i are C i -quasiconvex in (G, d i+1 ).
Take A and B two distinct elements of C H 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 × B, with b i on that quasigeodesic, and such that a −1 i b i is an element of length at most 20δ 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 (200δ i+1 )-far from one another. Apply the Proposition 2.12 (or alternatively 2.15) 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 (200δC 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 (200δC h ) in the metric d), and a pair (a i0 , b i0 ), such that a −1 i b i = g 0 and such that for 1000δ 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 H i , corresponding to the cosets A and B, thus contains a −1 i0 a j for all those indices j. There are indices j for which a −1 i0 a j labels a quasi-geodesic paths in A of length at least 1000δ i+1 C i . Such an element is either loxodromic, or elliptic with fixed point at the midpoint [a i0 , 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 −1 i0 a j1 a −1 i0 a j2 is loxodromic. This element is in the intersection of conjugates of elements of H i , thus is in a subgroup among the conjugates of some H j for j i + 1, but therefore must have diameter 2 in the metric d i+1 , and cannot contain loxodromic elements. This is thus a contradiction.
Cases 3 and 4: G = Mod(Σ) or Out(F n ), H convex cocompact. -Consider the Teichmüller metric on Teichmüller space (Teich(Σ), d T ) and the (symmetrization of the) Lipschitz metric on Outer space (cv n , d S ) respectively for Mod(Σ) and Out(F n ). Though Teich(Σ) and cv n are non-hyperbolic, they are proper metric spaces.
For the mapping class group Mod(Σ), the curve complex (CC Σ), d) is hyperbolic and quasi-isometric to (Mod(Σ), d), where d is the word-metric on Mod(Σ) obtained by taking as generating set a finite generating set of Mod(Σ) along with all elements of certain sub-mapping class groups (see [MM99]). By Definition 3.9 and Remark 3.11, for all n, all elements of C H n and of C H n+1 are C 1 -quasiconvex in (G, d). Therefore, by Proposition 2.11, all elements of C H 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 3.9, C H n is mutually cobounded in the metric d n+1 . Proposition 2.10 now shows that the horoballification of (G, d n+1 ) over C H n is hyperbolic, for all n. Proposition 2.23 then guarantees that C H 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.

Graded relative hyperbolicity for quasiconvex subgroups
Proposition 5.2. -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.
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 1.3).
By Theorem 3.2, H has finite height. By Proposition 4.15 it satisfies the uniform qiintersection property 3.9. Therefore, by Proposition 5.1, 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.
Proposition 5.3. -Let (G, P) 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 P) and let d be the word metric with respect to S ∪ P. Then (G, {H}, d) has graded relative hyperbolicity as well as graded geometric relative hyperbolicity.
Proof. -The proof is similar to that of Proposition 5.2. By Theorem 4.6, H has finite relative height, hence it has finite geometric height for the relative metric (see Example 4.7).
Next, by Proposition 4.15, H satisfies the uniform qi-intersection property for a relative metric, and graded geometric relative hyperbolicity follows from Proposition 5.1.
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.
Similarly, replacing the use of Theorem 3.2 by Theorems 4.10 and 4.14, one obtains the following.
Proposition 5.4. -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 CC(S) (respectively the free factor complex F n ). Let H be a convex cocompact subgroup of G. Then (G, {H}, d) has graded relative hyperbolicity. the pair of meeting points is in an item of H i+1 inside H i, , or Y ⊂ H i, . In both cases, by the inductive assumption, there is a path in H +ε∆ i, between the meeting points in H i, that is a quasigeodesic for d. Hence the second assumption of Proposition 2.12 is satisfied.
We can thus conclude by that proposition that H i, 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, i.e., quasiconvexity follows from graded geometric relative hyperbolicity.
We shall deduce various consequences of Proposition 6.1 below. However, before we proceed, we need the following observation since we are dealing with spaces/graphs that are not necessarily proper.
Observation 6.2. -Let X be a (not necessarily proper) hyperbolic graph. For all C 0 0, there exists C 1 0 such that the following holds: Let H be a hyperbolic group acting uniformly properly on X, i.e., for all D 0 there exists N such that for any x ∈ 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.
Combining Proposition 6.1 with Observation 6.2 we obtain the following: Then H is hyperbolic and H is qi-embedded in (G, d).
Proof. -Quasi-convexity of H in (G, d) was established in Proposition 6.1. Qi-embeddedness of H follows from Observation 6.2. Hyperbolicity of H is an immediate consequence.
6.2. The main theorem. -We assemble the pieces now to prove the following main theorem of the paper. (1) G a hyperbolic group and d the word metric with respect to a finite generating set S.
(2) G is finitely generated and hyperbolic relative to P, S a finite relative generating set, and d the word metric with respect to S ∪ P.
(3) 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 quasiisometric to the curve complex CC(S).
(4) 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 F n . Proof. -The forward implications of quasiconvexity to graded geometric relative hyperbolicity in the first 3 cases are proved by Propositions 5.2, 5.3, 5.4 and 5.5 and case 4 by Proposition 5.4. 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 6.1.
The proofs of the reverse implications of (3) and (4) are similar. Proposition 6.3 proves that any orbit of H on either the curve complex CC(S) or the free factor complex F n is qi-embedded. Convex cocompactness now follows from Theorems 4.9 and 4.13.
6.3. Examples. -We give a couple of examples below to show that finiteness of geometric height does not necessarily follow from quasiconvexity. Example 6.5. -Let G 1 = π 1 (S) and H = h be a cyclic subgroup corresponding to a simple closed curve. Let G 2 = H 1 ⊕ H 2 , where each H i is isomorphic to Z. Let G = G 1 * H=H1 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.
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.
Example 6.6. -Let G = a i , b i : i ∈ N, a bi 2i = a 2i−1 and let F be the (free) subgroup generated by {a i }. Then F bi ∩ F = a 2i−1 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 bi ∩ F is infinite.