logo Journal de l'École Polytechnique
Height, graded relative hyperbolicity and quasiconvexity
[Hauteur, hyperbolicité relative graduée, et quasiconvexité]
François Dahmani1 ; Mahan Mj2
1 Université Grenoble Alpes, Institut Fourier F-38000 Grenoble, France
2 Tata Institute of Fundamental Research 1, Homi Bhabha Road, Mumbai-400005, India
Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 515-556.

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.

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 ).

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/jep.50
Classification : 20F65, 20F67, 22E40
Keywords: Quasiconvex subgroups, hyperbolic groups, relatively hyperbolic groups, height, convex cocompact subgroups
Mot clés : Sous-groupes quasi-convexes, groupes hyperboliques, groupes relativement hyperboliques, groupes convexes cocompacts
François Dahmani 1 ; Mahan Mj 2

1 Université Grenoble Alpes, Institut Fourier F-38000 Grenoble, France
2 Tata Institute of Fundamental Research 1, Homi Bhabha Road, Mumbai-400005, India
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{JEP_2017__4__515_0,
     author = {Fran\c{c}ois Dahmani and Mahan Mj},
     title = {Height, graded relative hyperbolicity and quasiconvexity},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {515--556},
     publisher = {\'Ecole polytechnique},
     volume = {4},
     year = {2017},
     doi = {10.5802/jep.50},
     zbl = {06754335},
     mrnumber = {3646028},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.50/}
}
TY  - JOUR
AU  - François Dahmani
AU  - Mahan Mj
TI  - Height, graded relative hyperbolicity and quasiconvexity
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2017
SP  - 515
EP  - 556
VL  - 4
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.50/
DO  - 10.5802/jep.50
LA  - en
ID  - JEP_2017__4__515_0
ER  - 
%0 Journal Article
%A François Dahmani
%A Mahan Mj
%T Height, graded relative hyperbolicity and quasiconvexity
%J Journal de l’École polytechnique — Mathématiques
%D 2017
%P 515-556
%V 4
%I École polytechnique
%U https://jep.centre-mersenne.org/articles/10.5802/jep.50/
%R 10.5802/jep.50
%G en
%F JEP_2017__4__515_0
François Dahmani; Mahan Mj. Height, graded relative hyperbolicity and quasiconvexity. Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 515-556. doi : 10.5802/jep.50. https://jep.centre-mersenne.org/articles/10.5802/jep.50/

[Bes04] M. Bestvina - “Geometric group theory problem list” (2004), http:math.utah.edu/~bestvina

[BF14] M. Bestvina & M. Feighn - “Hyperbolicity of the complex of free factors”, Adv. in Math. 256 (2014), p. 104-155, Corrigendum: Ibid., 259 (2014), p. 843 | DOI | MR | Zbl

[Bow12] B. H. Bowditch - “Relatively hyperbolic groups”, Internat. J. Algebra Comput. 22 (2012) no. 3, 1250016, 66 pages | DOI | MR | Zbl

[Cou14] R. Coulon - “On the geometry of Burnside quotients of torsion free hyperbolic groups”, Internat. J. Algebra Comput. 24 (2014) no. 3, p. 251-345 | DOI | MR | Zbl

[Dah03] F. Dahmani - “Combination of convergence groups”, Geom. Topol. 7 (2003), p. 933-963 | DOI | MR | Zbl

[DDM14] S. Dowdall, M. Duchin & H. Masur - “Statistical hyperbolicity in Teichmüller space”, Geom. Funct. Anal. 24 (2014) no. 3, p. 748-795 | DOI | Zbl

[DGO17] F. Dahmani, V. Guirardel & D. Osin - Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces, Mem. Amer. Math. Soc., vol. 245, no.  1156, American Mathematical Society, Providence, RI, 2017 | Zbl

[DH15] F. Dahmani & C. Horbez - “Spectral theorems for random walks on mapping class groups and Out(F N )” (2015), arXiv:1506.06790

[DM15] S. Das & M. Mj - “Controlled Floyd separation and non relatively hyperbolic groups”, J. Ramanujan Math. Soc. 30 (2015) no. 3, p. 267-294 | MR

[DS05] C. Druţu & M. Sapir - “Tree-graded spaces and asymptotic cones of groups”, Topology 44 (2005) no. 5, p. 959-1058, With an appendix by D. Osin and M. Sapir | DOI | MR | Zbl

[DT14] S. Dowdall & S. J. Taylor - “Hyperbolic extensions of free groups” (2014), arXiv:1406.2567

[DT15] M. G. Durham & S. J. Taylor - “Convex cocompactness and stability in mapping class groups”, Algebraic Geom. Topol. 15 (2015) no. 5, p. 2839-2859 | DOI | MR | Zbl

[Far98] B. Farb - “Relatively hyperbolic groups”, Geom. Funct. Anal. 8 (1998) no. 5, p. 810-840 | DOI | MR | Zbl

[FM02] B. Farb & L. Mosher - “Convex cocompact subgroups of mapping class groups”, Geom. Topol. 6 (2002), p. 91-152 | DOI | MR | Zbl

[GM08] D. Groves & J. F. Manning - “Dehn filling in relatively hyperbolic groups”, Israel J. Math. 168 (2008), p. 317-429 | DOI | MR | Zbl

[GMRS98] R. Gitik, M. Mitra, E. Rips & M. Sageev - “Widths of subgroups”, Trans. Amer. Math. Soc. 350 (1998) no. 1, p. 321-329 | DOI | MR | Zbl

[Ham08] U. Hamenstädt - “Word hyperbolic extensions of surface groups” (2008), arXiv:0807.4891v2

[Ham10] U. Hamenstädt - “Stability of quasi-geodesics in Teichmüller space”, Geom. Dedicata 146 (2010), p. 101-116 | DOI | Zbl

[Hru10] G. C. Hruska - “Relative hyperbolicity and relative quasiconvexity for countable groups”, Algebraic Geom. Topol. 10 (2010) no. 3, p. 1807-1856 | DOI | MR | Zbl

[HW09] G. C. Hruska & D. T. Wise - “Packing subgroups in relatively hyperbolic groups”, Geom. Topol. 13 (2009) no. 4, p. 1945-1988 | DOI | MR | Zbl

[KL08] R. P. Kent & C. J. Leininger - “Shadows of mapping class groups: capturing convex cocompactness”, Geom. Funct. Anal. 18 (2008) no. 4, p. 1270-1325 | DOI | MR | Zbl

[Kla99] E. Klarreich - “Semiconjugacies between Kleinian group actions on the Riemann sphere”, Amer. J. Math. 121 (1999) no. 5, p. 1031-1078 | DOI | MR | Zbl

[Mas80] H. Masur - “Uniquely ergodic quadratic differentials”, Comment. Math. Helv. 55 (1980) no. 2, p. 255-266 | DOI | MR | Zbl

[McM01] C. T. McMullen - “Local connectivity, Kleinian groups and geodesics on the blowup of the torus”, Invent. Math. 146 (2001) no. 1, p. 35-91 | DOI | MR | Zbl

[Mj08] M. Mj - “Relative rigidity, quasiconvexity and C-complexes”, Algebraic Geom. Topol. 8 (2008) no. 3, p. 1691-1716 | DOI | MR | Zbl

[Mj10] M. Mj - “Cannon-Thurston maps, i-bounded geometry and a theorem of McMullen”, in Sémin. Théor. Spectr. Géom., vol. 28, Univ. Grenoble I, Saint-Martin-d’Hères, 2010, p. 63-107 | Numdam | MR | Zbl

[Mj14] M. Mj - “Cannon-Thurston maps for surface groups”, Ann. of Math. (2) 179 (2014) no. 1, p. 1-80 | DOI | MR | Zbl

[MM99] H. A. Masur & Y. N. Minsky - “Geometry of the complex of curves. I. Hyperbolicity”, Invent. Math. 138 (1999) no. 1, p. 103-149 | DOI | MR | Zbl

[Osi06a] D. V. Osin - “Elementary subgroups of relatively hyperbolic groups and bounded generation”, Internat. J. Algebra Comput. 16 (2006) no. 1, p. 99-118 | DOI | MR | Zbl

[Osi06b] D. V. Osin - Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems, Mem. Amer. Math. Soc., vol. 179, no.  843, American Mathematical Society, Providence, RI, 2006 | Zbl

[Sho91] H. Short - “Quasiconvexity and a theorem of Howson’s”, in Group theory from a geometrical viewpoint (Trieste, 1990), World Sci. Publ., River Edge, NJ, 1991, p. 168-176 | MR | Zbl

[Szc98] A. Szczepański - “Relatively hyperbolic groups”, Michigan Math. J. 45 (1998) no. 3, p. 611-618 | DOI | MR | Zbl

Cité par Sources :