Height, graded relative hyperbolicity and quasiconvexity
[Hauteur, hyperbolicité relative graduée, et quasiconvexité]
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 :
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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
Dahmani, François 1 ; Mj, Mahan 2

1 Université Grenoble Alpes, Institut Fourier F-38000 Grenoble, France
2 Tata Institute of Fundamental Research 1, Homi Bhabha Road, Mumbai-400005, India
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Dahmani, François; Mj, Mahan. Height, graded relative hyperbolicity and quasiconvexity. Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 515-556. doi : 10.5802/jep.50. http://www.numdam.org/articles/10.5802/jep.50/

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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