A non-residually finite group acting uniformly properly on a hyperbolic space
[Un exemple de groupe non résiduellement fini muni d’une action uniformément propre sur un espace hyperbolique]
Coulon, Rémi1 ; Osin, Denis2
1 Université de Rennes, CNRS, IRMAR - UMR 6625 Campus de Beaulieu, 263 avenue du Général Leclerc, F-35000 Rennes, France
2 Department of Mathematics, Vanderbilt University Nashville, TN 37240, U.S.A.
Journal de l’École polytechnique — Mathématiques, Tome 6 (2019), pp. 19-30.

Dans cet article nous construisons un exemple de groupe qui n’est pas résiduellement fini et qui est muni d’une action uniformément propre sur un espace hyperbolique au sens de Gromov.

In this article we produce an example of a non-residually finite group which admits a uniformly proper action on a Gromov hyperbolic space.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/jep.86
Classification : 20F65, 20F67, 20E26, 20F06
Keywords: Hyperbolic spaces, residually finite group, small cancellation theory, uniformly proper action, bounded geometry
Mot clés : Espaces hyperboliques, groupes résiduellement finis, théorie de la petite simplification, action uniformément propre, géométrie bornée
Coulon, Rémi 1 ; Osin, Denis 2

1 Université de Rennes, CNRS, IRMAR - UMR 6625 Campus de Beaulieu, 263 avenue du Général Leclerc, F-35000 Rennes, France
2 Department of Mathematics, Vanderbilt University Nashville, TN 37240, U.S.A. 
@article{JEP_2019__6__19_0,
     author = {Coulon, R\'emi and Osin, Denis},
     title = {A non-residually finite group acting uniformly properly on a hyperbolic space},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {19--30},
     publisher = {Ecole polytechnique},
     volume = {6},
     year = {2019},
     doi = {10.5802/jep.86},
     zbl = {07003360},
     mrnumber = {3896034},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jep.86/}
}
TY  - JOUR
AU  - Coulon, Rémi
AU  - Osin, Denis
TI  - A non-residually finite group acting uniformly properly on a hyperbolic space
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2019
SP  - 19
EP  - 30
VL  - 6
PB  - Ecole polytechnique
UR  - http://www.numdam.org/articles/10.5802/jep.86/
DO  - 10.5802/jep.86
LA  - en
ID  - JEP_2019__6__19_0
ER  - 
%0 Journal Article
%A Coulon, Rémi
%A Osin, Denis
%T A non-residually finite group acting uniformly properly on a hyperbolic space
%J Journal de l’École polytechnique — Mathématiques
%D 2019
%P 19-30
%V 6
%I Ecole polytechnique
%U http://www.numdam.org/articles/10.5802/jep.86/
%R 10.5802/jep.86
%G en
%F JEP_2019__6__19_0
Coulon, Rémi; Osin, Denis. A non-residually finite group acting uniformly properly on a hyperbolic space. Journal de l’École polytechnique — Mathématiques, Tome 6 (2019), pp. 19-30. doi : 10.5802/jep.86. http://www.numdam.org/articles/10.5802/jep.86/

[1] Belegradek, I.; Osin, D. V. Rips construction and Kazhdan property (T), Groups Geom. Dyn., Volume 2 (2008) no. 1, pp. 1-12 | DOI | MR | Zbl

[2] Bowditch, B. H Tight geodesics in the curve complex, Invent. Math., Volume 171 (2008) no. 2, pp. 281-300 | DOI | MR | Zbl

[3] Bridson, M. R.; Haefliger, A. Metric spaces of non-positive curvature, Grundlehren Math. Wiss., 319, Springer-Verlag, Berlin, 1999 | MR | Zbl

[4] Coornaert, M.; Delzant, T.; Papadopoulos, A. Géométrie et théorie des groupes. Les groupes hyperboliques de Gromov, Lect. Notes in Math., 1441, Springer-Verlag, Berlin, 1990 | Zbl

[5] Coulon, R. Asphericity and small cancellation theory for rotation families of groups, Groups Geom. Dyn., Volume 5 (2011) no. 4, pp. 729-765 | DOI | MR | Zbl

[6] 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

[7] Coulon, R. Partial periodic quotients of groups acting on a hyperbolic space, Ann. Inst. Fourier (Grenoble), Volume 66 (2016) no. 5, pp. 1773-1857 | DOI | MR

[8] Dahmani, F.; Guirardel, V.; Osin, D. V. 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

[9] Delzant, T.; Gromov, M. Courbure mésoscopique et théorie de la toute petite simplification, J. Topology, Volume 1 (2008) no. 4, pp. 804-836 | DOI | Zbl

[10] Sur les groupes hyperboliques d’après Mikhael Gromov (Ghys, É.; de la Harpe, P., eds.), Progress in Math., 83, Birkhäuser Boston, Inc., Boston, MA, 1990 | Zbl

[11] Gromov, M. Hyperbolic groups, Essays in group theory, Springer, New York, 1987, pp. 75-263 | DOI | Zbl

[12] Gromov, M. Mesoscopic curvature and hyperbolicity, Contemporary Mathematics, 288, American Mathematical Society, Providence, RI, 2001 | MR | Zbl

[13] Groves, D.; Manning, J. F. Dehn filling in relatively hyperbolic groups, Israel J. Math., Volume 168 (2008) no. 1, pp. 317-429 | DOI | MR | Zbl

[14] Martino, A.; Minasyan, A. Conjugacy in normal subgroups of hyperbolic groups, Forum Math., Volume 24 (2012) no. 5, pp. 889-910 | DOI | MR | Zbl

[15] Ol’shanskii, A. Yu On residualing homomorphisms and G-subgroups of hyperbolic groups, Internat. J. Algebra Comput., Volume 3 (1993) no. 4, pp. 365-409 | DOI | MR | Zbl

[16] Osin, D. V. Acylindrically hyperbolic groups, Trans. Amer. Math. Soc., Volume 368 (2016) no. 2, pp. 851-888 | DOI | MR | Zbl

[17] Serre, J.-P. Arbres, amalgames, SL 2 , Astérisque, 46, Société Mathématique de France, Paris, 1977 | Zbl

Cité par Sources :