Infinitely presented graphical small cancellation groups are acylindrically hyperbolic

Gruber, Dominik; Sisto, Alessandro

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 $Gr\left(7\right)$ sont acylindriquement hyperboliques. Cette classe contient les groupes satisfaisant la condition classique de petite simplification graphique $C\left(7\right)$ et par conséquent ceux vérifiant la condition $C^{\text{'}}\left(\frac{1}{6}\right)$. 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 $C^{\text{'}}\left(\frac{1}{6}\right)$ qui fournissent de nouveaux exemples de fonctions de divergence des groupes.

We prove that infinitely presented graphical $Gr\left(7\right)$ small cancellation groups are acylindrically hyperbolic. In particular, infinitely presented classical $C\left(7\right)$-groups and, hence, classical $C^{\text{'}}\left(\frac{1}{6}\right)$-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 $C^{\text{'}}\left(\frac{1}{6}\right)$-groups that provide new examples of divergence functions of groups.

Classification : 20F06, 20F65, 20F67
Keywords: Graphical small cancellation, acylindrical hyperbolicity, divergence
Mot clés : Petite simplification graphique, hyperbolicité acylindrique, divergence

[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 $\mathrm{CAT}\left(0\right)$ 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 $C\left(6\right)$ and $C\left(7\right)$ small cancellation presentations, Trans. Am. Math. Soc., Volume 367 (2015) no. 3, pp. 2051-2078 | Zbl

[27] Gruber, Dominik Infinitely presented $C\left(6\right)$-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