Nous démontrons que les groupes de présentation infinie satisfaisant la condition de petite simplification graphique sont acylindriquement hyperboliques. Cette classe contient les groupes satisfaisant la condition classique de petite simplification graphique et par conséquent ceux vérifiant la condition . 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 qui fournissent de nouveaux exemples de fonctions de divergence des groupes.
We prove that infinitely presented graphical small cancellation groups are acylindrically hyperbolic. In particular, infinitely presented classical -groups and, hence, classical -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 -groups that provide new examples of divergence functions of groups.
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Keywords: Graphical small cancellation, acylindrical hyperbolicity, divergence
Mot clés : Petite simplification graphique, hyperbolicité acylindrique, divergence
Gruber, Dominik 1 ; Sisto, Alessandro 1
@article{AIF_2018__68_6_2501_0, author = {Gruber, Dominik and Sisto, Alessandro}, title = {Infinitely presented graphical small cancellation groups are acylindrically hyperbolic}, journal = {Annales de l'Institut Fourier}, pages = {2501--2552}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {68}, number = {6}, year = {2018}, doi = {10.5802/aif.3215}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3215/} }
TY - JOUR AU - Gruber, Dominik AU - Sisto, Alessandro TI - Infinitely presented graphical small cancellation groups are acylindrically hyperbolic JO - Annales de l'Institut Fourier PY - 2018 SP - 2501 EP - 2552 VL - 68 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3215/ DO - 10.5802/aif.3215 LA - en ID - AIF_2018__68_6_2501_0 ER -
%0 Journal Article %A Gruber, Dominik %A Sisto, Alessandro %T Infinitely presented graphical small cancellation groups are acylindrically hyperbolic %J Annales de l'Institut Fourier %D 2018 %P 2501-2552 %V 68 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3215/ %R 10.5802/aif.3215 %G en %F AIF_2018__68_6_2501_0
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. https://aif.centre-mersenne.org/articles/10.5802/aif.3215/
[1] Commensurating endomorphisms of acylindrically hyperbolic groups and applications, Groups Geom. Dyn., Volume 10 (2016) no. 4, pp. 1149-1210 | DOI | MR | Zbl
[2] Negative curvature in graphical small cancellation groups (2016) (to appear in Groups Geom. Dyn., http://arxiv.org/abs/1602.03767)
[3] Examples of random groups (2008) (www.mat.univie.ac.at/ arjantseva/Abs/random.pdf)
[4] Geometry of infinitely presented small cancellation groups, Rapid Decay and quasi-homomorphisms (2012) (http://arxiv.org/abs/1212.5280)
[5] Acylindrical hyperbolicity of cubical small-cancellation groups (2016) (http://arxiv.org/abs/1603.05725)
[6] Graphical small cancellation groups with the Haagerup property (2014) (http://arxiv.org/abs/1404.6807)
[7] Rips construction without unique product (2014) (http://arxiv.org/abs/arXiv:1407.2441)
[8] Asymptotic geometry of the mapping class group and Teichmüller space, Geom. Topol., Volume 10 (2006), pp. 1523-1578 | DOI | Zbl
[9] Divergence and quasimorphisms of right-angled Artin groups, Math. Ann., Volume 352 (2012) no. 2, pp. 339-356 | DOI | Zbl
[10] 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] Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity, Math. Ann., Volume 344 (2009) no. 3, pp. 543-595 | DOI | Zbl
[12] Bounded cohomology of subgroups of mapping class groups, Geom. Topol., Volume 6 (2002), pp. 69-89 | DOI | Zbl
[13] 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] Continuously many quasi-isometry classes of -generator groups, Comment. Math. Helv., Volume 73 (1998) no. 2, pp. 232-236 | DOI | Zbl
[15] Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, 319, Springer, 1999, xxii+643 pages | Zbl
[16] Small cancellation theory over Burnside groups (2017) (http://arxiv.org/abs/1705.09651)
[17] 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] Relatively hyperbolic groups: geometry and quasi-isometric invariance, Comment. Math. Helv., Volume 84 (2009) no. 3, pp. 503-546 | DOI | Zbl
[19] 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] 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] 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] Extending higher-dimensional quasi-cocycles, J. Topol., Volume 8 (2015) no. 4, pp. 1123-1155 | DOI | MR | Zbl
[23] Quadratic divergence of geodesics in spaces, Geom. Funct. Anal., Volume 4 (1994) no. 1, pp. 37-51 | DOI | Zbl
[24] Asymptotic invariants of infinite groups, London Mathematical Society Lecture Note Series, 182, Cambridge University Press, 1993, pp. 1-295 | Zbl
[25] Random walk in random groups, Geom. Funct. Anal., Volume 13 (2003) no. 1, pp. 73-146 | DOI | Zbl
[26] Groups with graphical and small cancellation presentations, Trans. Am. Math. Soc., Volume 367 (2015) no. 3, pp. 2051-2078 | Zbl
[27] Infinitely presented -groups are SQ-universal, J. Lond. Math. Soc., Volume 92 (2015) no. 1, pp. 178-201 | Zbl
[28] Infinitely presented graphical small cancellation groups, University of Vienna (Austria) (2015) (Ph. D. Thesis)
[29] Gromov’s random monsters do not act non-elementarily on hyperbolic spaces (2017) (http://arxiv.org/abs/1705.10258)
[30] Bounded cohomology and isometry groups of hyperbolic spaces, J. Eur. Math. Soc., Volume 10 (2008) no. 2, pp. 315-349 | DOI | Zbl
[31] Counterexamples to the Baum-Connes conjecture, Geom. Funct. Anal., Volume 12 (2002) no. 2, pp. 330-354 | DOI | Zbl
[32] Induced quasicocycles on groups with hyperbolically embedded subgroups, Algebr. Geom. Topol., Volume 13 (2013) no. 5, pp. 2635-2665 | DOI | Zbl
[33] Combinatorial group theory, Springer, 1977, xiv+339 pages (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89)
[34] CAT(0) spaces with polynomial divergence of geodesics, Geom. Dedicata, Volume 163 (2013), pp. 361-378 | DOI | MR | Zbl
[35] Acylindrical hyperbolicity of groups acting on trees, Math. Ann., Volume 362 (2015) no. 3-4, pp. 1055-1105 | DOI | MR | Zbl
[36] 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] 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] Small cancellation labellings of some infinite graphs and applications (2014) (http://arxiv.org/abs/1406.5015)
[39] Elementary subgroups of relatively hyperbolic groups and bounded generation, Int. J. Algebra Comput., Volume 16 (2006) no. 1, pp. 99-118 | DOI | Zbl
[40] Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems, Mem. Am. Math. Soc., Volume 179 (2006) no. 843, vi+100 pages | Zbl
[41] Acylindrically hyperbolic groups, Trans. Am. Math. Soc., Volume 368 (2016) no. 2, pp. 851-888 | DOI | MR | Zbl
[42] Strongly geodesically automatic groups are hyperbolic, Invent. Math., Volume 121 (1995) no. 2, pp. 323-334 | DOI | MR | Zbl
[43] 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] Contracting elements and random walks (2011) (to appear in J. Reine Angew. Math., http://arxiv.org/abs/1112.2666)
[45] On metric relative hyperbolicity (2012) (http://arxiv.org/abs/1210.8081)
[46] Quasi-convexity of hyperbolically embedded subgroups, Math. Z., Volume 283 (2016) no. 3-4, pp. 649-658 | DOI | MR | Zbl
[47] Rips-Segev torsion-free groups without unique product, J. Algebra, Volume 438 (2015), pp. 337-378 | Zbl
[48] 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] Asymptotic cones of finitely generated groups, Bull. Lond. Math. Soc., Volume 32 (2000) no. 2, pp. 203-208 | DOI | Zbl
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