Infinitely presented graphical small cancellation groups are acylindrically hyperbolic
[Les groupes à petite simplification graphique de présentation infinie sont acylindriquement hyperboliques]
Annales de l'Institut Fourier, Tome 68 (2018) no. 6, pp. 2501-2552.

Nous démontrons que les groupes de présentation infinie satisfaisant la condition de petite simplification graphique Gr(7) sont acylindriquement hyperboliques. Cette classe contient les groupes satisfaisant la condition classique de petite simplification graphique C(7) et par conséquent ceux vérifiant la condition C ' (1 6). Plus généralement, nous démontrons des énoncés analogues valables pour les presentations à petite simplification graphique dans un produit libre. Nous construisons des présentations infinies vérifiant la conditions classique C ' (1 6) qui fournissent de nouveaux exemples de fonctions de divergence des groupes.

We prove that infinitely presented graphical Gr(7) small cancellation groups are acylindrically hyperbolic. In particular, infinitely presented classical C(7)-groups and, hence, classical C ' (1 6)-groups are acylindrically hyperbolic. We also prove the analogous statements for the larger class of graphical small cancellation presentations over free products. We construct infinitely presented classical C ' (1 6)-groups that provide new examples of divergence functions of groups.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3215
Classification : 20F06, 20F65, 20F67
Keywords: Graphical small cancellation, acylindrical hyperbolicity, divergence
Mot clés : Petite simplification graphique, hyperbolicité acylindrique, divergence
Gruber, Dominik 1 ; Sisto, Alessandro 1

1 Department of Mathematics, ETH Zurich 8092 Zurich (Switzerland)
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Gruber, Dominik; Sisto, Alessandro. Infinitely presented graphical small cancellation groups are acylindrically hyperbolic. Annales de l'Institut Fourier, Tome 68 (2018) no. 6, pp. 2501-2552. doi : 10.5802/aif.3215. http://www.numdam.org/articles/10.5802/aif.3215/

[1] Antolín, Yago; Minasyan, Ashot; Sisto, Alessandro Commensurating endomorphisms of acylindrically hyperbolic groups and applications, Groups Geom. Dyn., Volume 10 (2016) no. 4, pp. 1149-1210 | DOI | MR | Zbl

[2] Arzhantseva, Goulnara; Cashen, Christopher H.; Gruber, Dominik; Hume, David Negative curvature in graphical small cancellation groups (2016) (to appear in Groups Geom. Dyn., http://arxiv.org/abs/1602.03767)

[3] Arzhantseva, Goulnara; Delzant, Thomas Examples of random groups (2008) (www.mat.univie.ac.at/ arjantseva/Abs/random.pdf)

[4] Arzhantseva, Goulnara; Druţu, Cornelia Geometry of infinitely presented small cancellation groups, Rapid Decay and quasi-homomorphisms (2012) (http://arxiv.org/abs/1212.5280)

[5] Arzhantseva, Goulnara; Hagen, Mark F. Acylindrical hyperbolicity of cubical small-cancellation groups (2016) (http://arxiv.org/abs/1603.05725)

[6] Arzhantseva, Goulnara; Osajda, Damian Graphical small cancellation groups with the Haagerup property (2014) (http://arxiv.org/abs/1404.6807)

[7] Arzhantseva, Goulnara; Steenbock, Markus Rips construction without unique product (2014) (http://arxiv.org/abs/arXiv:1407.2441)

[8] Behrstock, Jason Asymptotic geometry of the mapping class group and Teichmüller space, Geom. Topol., Volume 10 (2006), pp. 1523-1578 | DOI | Zbl

[9] Behrstock, Jason; Charney, Ruth Divergence and quasimorphisms of right-angled Artin groups, Math. Ann., Volume 352 (2012) no. 2, pp. 339-356 | DOI | Zbl

[10] Behrstock, Jason; Druţu, Cornelia Divergence, thick groups, and short conjugators, Ill. J. Math., Volume 58 (2014) no. 4, pp. 939-980 http://projecteuclid.org/euclid.ijm/1446819294 | MR | Zbl

[11] Behrstock, Jason; Druţu, Cornelia; Mosher, Lee Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity, Math. Ann., Volume 344 (2009) no. 3, pp. 543-595 | DOI | Zbl

[12] Bestvina, Mladen; Fujiwara, Koji Bounded cohomology of subgroups of mapping class groups, Geom. Topol., Volume 6 (2002), pp. 69-89 | DOI | Zbl

[13] Bowditch, Brian H. A short proof that a subquadratic isoperimetric inequality implies a linear one, Mich. Math. J., Volume 42 (1995) no. 1, pp. 103-107 | DOI | MR | Zbl

[14] Bowditch, Brian H. Continuously many quasi-isometry classes of 2-generator groups, Comment. Math. Helv., Volume 73 (1998) no. 2, pp. 232-236 | DOI | Zbl

[15] Bridson, Martin R.; Haefliger, André Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, 319, Springer, 1999, xxii+643 pages | Zbl

[16] Coulon, Rémi; Gruber, Dominik Small cancellation theory over Burnside groups (2017) (http://arxiv.org/abs/1705.09651)

[17] Dahmani, François; Guirardel, Vincent; Osin, Denis Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces, Mem. Am. Math. Soc., Volume 245 (2017) no. 1156, v+152 pages | DOI | MR

[18] Druţu, Cornelia Relatively hyperbolic groups: geometry and quasi-isometric invariance, Comment. Math. Helv., Volume 84 (2009) no. 3, pp. 503-546 | DOI | Zbl

[19] Druţu, Cornelia; Mozes, Shahar; Sapir, Mark Divergence in lattices in semisimple Lie groups and graphs of groups, Trans. Am. Math. Soc., Volume 362 (2010) no. 5, pp. 2451-2505 corrigendum in ibid. 370 (2018), no. 1, p. 749-754 | DOI | Zbl

[20] 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 D. Osin and Sapir) | Zbl

[21] Duchin, Moon; Rafi, Kasra Divergence of geodesics in Teichmüller space and the mapping class group, Geom. Funct. Anal., Volume 19 (2009) no. 3, pp. 722-742 | DOI | Zbl

[22] Frigerio, Roberto; Pozzetti, Maria B.; Sisto, Alessandro Extending higher-dimensional quasi-cocycles, J. Topol., Volume 8 (2015) no. 4, pp. 1123-1155 | DOI | MR | Zbl

[23] Gersten, Stephen M. Quadratic divergence of geodesics in CAT (0) spaces, Geom. Funct. Anal., Volume 4 (1994) no. 1, pp. 37-51 | DOI | Zbl

[24] Gromov, Mikhael Asymptotic invariants of infinite groups, London Mathematical Society Lecture Note Series, 182, Cambridge University Press, 1993, pp. 1-295 | Zbl

[25] Gromov, Mikhael Random walk in random groups, Geom. Funct. Anal., Volume 13 (2003) no. 1, pp. 73-146 | DOI | Zbl

[26] Gruber, Dominik Groups with graphical C(6) and C(7) small cancellation presentations, Trans. Am. Math. Soc., Volume 367 (2015) no. 3, pp. 2051-2078 | Zbl

[27] Gruber, Dominik Infinitely presented C(6)-groups are SQ-universal, J. Lond. Math. Soc., Volume 92 (2015) no. 1, pp. 178-201 | Zbl

[28] Gruber, Dominik Infinitely presented graphical small cancellation groups, University of Vienna (Austria) (2015) (Ph. D. Thesis)

[29] Gruber, Dominik; Sisto, Alessandro; Tessera, Romain Gromov’s random monsters do not act non-elementarily on hyperbolic spaces (2017) (http://arxiv.org/abs/1705.10258)

[30] Hamenstädt, Ursula Bounded cohomology and isometry groups of hyperbolic spaces, J. Eur. Math. Soc., Volume 10 (2008) no. 2, pp. 315-349 | DOI | Zbl

[31] Higson, Nigel; Lafforgue, Vincent; Skandalis, Georges Counterexamples to the Baum-Connes conjecture, Geom. Funct. Anal., Volume 12 (2002) no. 2, pp. 330-354 | DOI | Zbl

[32] Hull, Michael; Osin, Denis Induced quasicocycles on groups with hyperbolically embedded subgroups, Algebr. Geom. Topol., Volume 13 (2013) no. 5, pp. 2635-2665 | DOI | Zbl

[33] Lyndon, Roger C.; Schupp, Paul E. Combinatorial group theory, Springer, 1977, xiv+339 pages (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89)

[34] Macura, Nataša CAT(0) spaces with polynomial divergence of geodesics, Geom. Dedicata, Volume 163 (2013), pp. 361-378 | DOI | MR | Zbl

[35] Minasyan, Ashot; Osin, Denis Acylindrical hyperbolicity of groups acting on trees, Math. Ann., Volume 362 (2015) no. 3-4, pp. 1055-1105 | DOI | MR | Zbl

[36] Ollivier, Yann On a small cancellation theorem of Gromov, Bull. Belg. Math. Soc. Simon Stevin, Volume 13 (2006) no. 1, pp. 75-89 http://projecteuclid.org/euclid.bbms/1148059334 | Zbl

[37] Ol’shanskii, Alexander; Osin, Denis; Sapir, Mark Lacunary hyperbolic groups, Geom. Topol., Volume 13 (2009) no. 4, pp. 2051-2140 (With an appendix by Michael Kapovich and Bruce Kleiner) | DOI | Zbl

[38] Osajda, Damian Small cancellation labellings of some infinite graphs and applications (2014) (http://arxiv.org/abs/1406.5015)

[39] Osin, Denis Elementary subgroups of relatively hyperbolic groups and bounded generation, Int. J. Algebra Comput., Volume 16 (2006) no. 1, pp. 99-118 | DOI | Zbl

[40] Osin, Denis Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems, Mem. Am. Math. Soc., Volume 179 (2006) no. 843, vi+100 pages | Zbl

[41] Osin, Denis Acylindrically hyperbolic groups, Trans. Am. Math. Soc., Volume 368 (2016) no. 2, pp. 851-888 | DOI | MR | Zbl

[42] Papasoglu, Panos Strongly geodesically automatic groups are hyperbolic, Invent. Math., Volume 121 (1995) no. 2, pp. 323-334 | DOI | MR | Zbl

[43] Pride, Stephen J. Some problems in combinatorial group theory, Groups—Korea 1988 (Pusan, 1988) (Lecture Notes in Math.), Volume 1398, Springer, 1989, pp. 146-155 | DOI | Zbl

[44] Sisto, Alessandro Contracting elements and random walks (2011) (to appear in J. Reine Angew. Math., http://arxiv.org/abs/1112.2666)

[45] Sisto, Alessandro On metric relative hyperbolicity (2012) (http://arxiv.org/abs/1210.8081)

[46] Sisto, Alessandro Quasi-convexity of hyperbolically embedded subgroups, Math. Z., Volume 283 (2016) no. 3-4, pp. 649-658 | DOI | MR | Zbl

[47] Steenbock, Markus Rips-Segev torsion-free groups without unique product, J. Algebra, Volume 438 (2015), pp. 337-378 | Zbl

[48] Strebel, Ralph Appendix. Small cancellation groups, Sur les groupes hyperboliques d’après Mikhael Gromov (Bern, 1988) (Progress in Mathematics), Volume 83, Birkhäuser, 1990, pp. 227-273 | Zbl

[49] Thomas, Simon; Velickovic, Boban Asymptotic cones of finitely generated groups, Bull. Lond. Math. Soc., Volume 32 (2000) no. 2, pp. 203-208 | DOI | Zbl

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