Partial periodic quotients of groups acting on a hyperbolic space  [ Quotient partiellement périodique de groupes agissant sur une espace hyperbolique ]
Annales de l'Institut Fourier, Tome 66 (2016) no. 5, pp. 1773-1857.

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.

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.

Reçu le : 2014-05-28
Accepté le : 2015-06-23
Publié le : 2016-07-27
DOI : https://doi.org/10.5802/aif.3050
Classification : 10X99,  14A12,  11L05
Mots clés : géométrie hyperbolique, groupes périodiques, théorie de la petite simplification, action acylindrique, groupe modulaire de surface.
@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'institut Fourier},
     volume = {66},
     number = {5},
     year = {2016},
     doi = {10.5802/aif.3050},
     language = {en},
     url = {www.numdam.org/item/AIF_2016__66_5_1773_0/}
}
Coulon, Rémi B. Partial periodic quotients of groups acting on a hyperbolic space. Annales de l'Institut Fourier, Tome 66 (2016) no. 5, pp. 1773-1857. doi : 10.5802/aif.3050. http://www.numdam.org/item/AIF_2016__66_5_1773_0/

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

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

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

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

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

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

[7] Coornaert, M.; Delzant, T.; Papadopoulos, A. Géométrie et théorie des groupes, Lecture Notes in Mathematics, Volume 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., Volume 5 (2011) no. 4, pp. 729-765 | Article

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

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

[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., Volume 1 (2008) no. 4, pp. 804-836 | Article

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

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

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

[17] Farb, Benson; Margalit, Dan A primer on mapping class groups, Princeton Mathematical Series, Volume 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., Volume 188 (1999) no. 2, pp. 251-274 | Article

[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., Volume 26 (2014) no. 2, pp. 337-355 | Article

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

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

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

[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.) Volume 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., Volume 10 (2010) no. 3, pp. 1807-1856 | Article

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

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

[28] Kirby, Rob Problems in low-dimensional topology, Geometric topology (Athens, GA, 1993) (AMS/IP Stud. Adv. Math.) Volume 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., Volume 138 (1999) no. 1, pp. 103-149 | Article

[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., Volume 32 (1968), p. 212-244, 251–524, 709–731

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

[33] Olʼshanskiĭ, A. Yu. Periodic quotient groups of hyperbolic groups, Mat. Sb., Volume 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, Volume 148, Springer-Verlag, New York, 1995, xvi+513 pages | Article

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

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

[38] Serre, Jean-Pierre Rigidité du foncteur de Jacobi d’échelon n3, 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., Volume 45 (1998) no. 3, pp. 611-618 | Article

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