no. 5
Partial periodic quotients of groups acting on a hyperbolic space
Annales de l'Institut Fourier, Volume 66 (2016) no. 5, pp. 1773-1857.

In this article, we construct partial periodic quotients of groups which have a non-elementary acylindrical action on a hyperbolic space. In particular, we provide infinite quotients of mapping class groups where a fixed power of every homeomorphism is identified with a periodic or reducible element.

Dans cet article, nous construisons des quotients partiellement périodiques de groupes admettant une action acylindrique sur un espace hyperbolique. En particulier, nous produisons des quotients infinis de groupes modulaires de surfaces, dans lesquelles une puissance fixée de tout homéomorphisme s’identifie avec un élément réductible ou un élément d’ordre fini.

Received:
Accepted:
Published online:
DOI: 10.5802/aif.3050
Classification: 10X99,  14A12,  11L05
Keywords: Small cancellation theory, mapping class groups, hyperbolic spaces, periodic quotients 
@article{AIF_2016__66_5_1773_0,
     author = {Coulon, R\'emi B.},
     title = {Partial periodic quotients of groups acting on a hyperbolic space},
     journal = {Annales de l'Institut Fourier},
     pages = {1773--1857},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {66},
     number = {5},
     year = {2016},
     doi = {10.5802/aif.3050},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3050/}
}
TY  - JOUR
TI  - Partial periodic quotients of groups acting on a hyperbolic space
JO  - Annales de l'Institut Fourier
PY  - 2016
DA  - 2016///
SP  - 1773
EP  - 1857
VL  - 66
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3050/
UR  - https://doi.org/10.5802/aif.3050
DO  - 10.5802/aif.3050
LA  - en
ID  - AIF_2016__66_5_1773_0
ER  - 
%0 Journal Article
%T Partial periodic quotients of groups acting on a hyperbolic space
%J Annales de l'Institut Fourier
%D 2016
%P 1773-1857
%V 66
%N 5
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.3050
%R 10.5802/aif.3050
%G en
%F AIF_2016__66_5_1773_0
Coulon, Rémi B. Partial periodic quotients of groups acting on a hyperbolic space. Annales de l'Institut Fourier, Volume 66 (2016) no. 5, pp. 1773-1857. doi : 10.5802/aif.3050. https://aif.centre-mersenne.org/articles/10.5802/aif.3050/

[1] Bestvina, Mladen; Feighn, Mark A hyperbolic Out (F n )-complex, Groups Geom. Dyn., Tome 4 (2010) no. 1, pp. 31-58 | DOI

[2] Bestvina, Mladen; Fujiwara, Koji Bounded cohomology of subgroups of mapping class groups, Geom. Topol., Tome 6 (2002), p. 69-89 (electronic) | DOI

[3] Bowditch, Brian H. Tight geodesics in the curve complex, Invent. Math., Tome 171 (2008) no. 2, pp. 281-300 | DOI

[4] Bowditch, Brian H. Relatively hyperbolic groups, Internat. J. Algebra Comput., Tome 22 (2012) no. 3, 1250016, 66 pages | DOI

[5] Bridson, Martin R.; Haefliger, André Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 319, Springer-Verlag, Berlin, 1999, xxii+643 pages | DOI

[6] Cantat, Serge; Lamy, Stéphane Normal subgroups in the Cremona group, Acta Math., Tome 210 (2013) no. 1, pp. 31-94 (With an appendix by Yves de Cornulier) | DOI

[7] Coornaert, M.; Delzant, T.; Papadopoulos, A. Géométrie et théorie des groupes, Lecture Notes in Mathematics, Tome 1441, Springer-Verlag, Berlin, 1990, x+165 pages (Les groupes hyperboliques de Gromov. [Gromov hyperbolic groups], With an English summary)

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

[9] Coulon, Rémi Outer automorphisms of free Burnside groups, Comment. Math. Helv., Tome 88 (2013) no. 4, pp. 789-811 | DOI

[10] Coulon, Rémi On the geometry of Burnside quotients of torsion free hyperbolic groups, Internat. J. Algebra Comput., Tome 24 (2014) no. 3, pp. 251-345 | DOI

[11] Dahmani, F; Guirardel, V; Osin, Denis Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces (http://arxiv.org/abs/1111.7048)

[12] Delzant, Thomas Sous-groupes à deux générateurs des groupes hyperboliques, Group theory from a geometrical viewpoint (Trieste, 1990), World Sci. Publ., River Edge, NJ, 1991, pp. 177-189

[13] Delzant, Thomas; Gromov, Misha Courbure mésoscopique et théorie de la toute petite simplification, J. Topol., Tome 1 (2008) no. 4, pp. 804-836 | DOI

[14] Druţu, Cornelia; Sapir, Mark Tree-graded spaces and asymptotic cones of groups, Topology, Tome 44 (2005) no. 5, pp. 959-1058 (With an appendix by Denis Osin and Sapir) | DOI

[15] Farb, Benson Relatively hyperbolic groups, Geom. Funct. Anal., Tome 8 (1998) no. 5, pp. 810-840 | DOI

[16] Farb, Benson Some problems on mapping class groups and moduli space, Problems on mapping class groups and related topics (Proc. Sympos. Pure Math.) Tome 74, Amer. Math. Soc., Providence, RI, 2006, pp. 11-55 | DOI

[17] Farb, Benson; Margalit, Dan A primer on mapping class groups, Princeton Mathematical Series, Tome 49, Princeton University Press, Princeton, NJ, 2012, xiv+472 pages

[18] Funar, Louis On the TQFT representations of the mapping class groups, Pacific J. Math., Tome 188 (1999) no. 2, pp. 251-274 | DOI

[19] Funar, Louis On power subgroups of mapping class groups (2009) (http://arxiv.org/abs/0910.1493)

[20] Funar, Louis; Kohno, Toshitake Free subgroups within the images of quantum representations, Forum Math., Tome 26 (2014) no. 2, pp. 337-355 | DOI

[21] Funar, Louis; Kohno, Toshitake On Burau’s representations at roots of unity, Geom. Dedicata, Tome 169 (2014), pp. 145-163 | DOI

[22] Sur les groupes hyperboliques d’après Mikhael Gromov (Ghys, É.; de la Harpe, P., eds.), Progress in Mathematics, Tome 83, Birkhäuser Boston, Inc., Boston, MA, 1990, xii+285 pages (Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988) | DOI

[23] Gromov, M. Hyperbolic groups, Essays in group theory (Math. Sci. Res. Inst. Publ.) Tome 8, Springer, New York, 1987, pp. 75-263 | DOI

[24] Harvey, W. J. Boundary structure of the modular group, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) (Ann. of Math. Stud.) Tome 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 245-251

[25] Hruska, G. Christopher Relative hyperbolicity and relative quasiconvexity for countable groups, Algebr. Geom. Topol., Tome 10 (2010) no. 3, pp. 1807-1856 | DOI

[26] Ivanov, Nikolai V. Fifteen problems about the mapping class groups, Problems on mapping class groups and related topics (Proc. Sympos. Pure Math.) Tome 74, Amer. Math. Soc., Providence, RI, 2006, pp. 71-80 | DOI

[27] Ivanov, S. V.; Olʼshanskiĭ, A. Yu. Hyperbolic groups and their quotients of bounded exponents, Trans. Amer. Math. Soc., Tome 348 (1996) no. 6, pp. 2091-2138 | DOI

[28] Kirby, Rob Problems in low-dimensional topology, Geometric topology (Athens, GA, 1993) (AMS/IP Stud. Adv. Math.) Tome 2, Amer. Math. Soc., Providence, RI, 1997, pp. 35-473

[29] Masur, Howard A.; Minsky, Yair N. Geometry of the complex of curves. I. Hyperbolicity, Invent. Math., Tome 138 (1999) no. 1, pp. 103-149 | DOI

[30] Minasyan, Ashot; Osin, Denis Acylindrical hyperbolicity of groups acting on trees (2013) (http://arxiv.org/abs/1310.6289)

[31] Novikov, P. S.; Adjan, S. I. Infinite periodic groups, Izv. Akad. Nauk SSSR Ser. Mat., Tome 32 (1968), p. 212-244, 251–524, 709–731

[32] Olʼshanskiĭ, A. Yu. The Novikov-Adyan theorem, Mat. Sb. (N.S.), Tome 118(160) (1982) no. 2, p. 203-235, 287

[33] Olʼshanskiĭ, A. Yu. Periodic quotient groups of hyperbolic groups, Mat. Sb., Tome 182 (1991) no. 4, pp. 543-567

[34] Osin, Denis Acylindrically hyperbolic groups (2013) (http://arxiv.org/abs/1304.1246)

[35] Rotman, Joseph J. An introduction to the theory of groups, Graduate Texts in Mathematics, Tome 148, Springer-Verlag, New York, 1995, xvi+513 pages | DOI

[36] Schur, I Über Gruppen periodischer linearer Substitutionen., Berl. Ber., Tome 1911 (1911), pp. 619-627

[37] Sela, Z. Acylindrical accessibility for groups, Invent. Math., Tome 129 (1997) no. 3, pp. 527-565 | DOI

[38] Serre, Jean-Pierre Rigidité du foncteur de Jacobi d’échelon n≥3, Séminaire Henri Cartan (1961), pp. 18-20

[39] Serre, Jean-Pierre Arbres, amalgames, SL 2 , Société Mathématique de France, Paris, 1977, 189 pp. (1 plate) pages (Avec un sommaire anglais, Rédigé avec la collaboration de Hyman Bass, Astérisque, No. 46)

[40] Sisto, Alessandro Contracting elements and random walks (2011) (http://arxiv.org/abs/1112.2666)

[41] Szczepański, Andrzej Relatively hyperbolic groups, Michigan Math. J., Tome 45 (1998) no. 3, pp. 611-618 | DOI

[42] Thurston, William P. On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.), Tome 19 (1988) no. 2, pp. 417-431 | DOI

Cited by Sources: