Hierarchically hyperbolic spaces, I: Curve complexes for cubical groups

Behrstock, Jason¹ ; Hagen, Mark² ; Sisto, Alessandro³

¹ The Graduate Center and Lehman College, CUNY, 365 5th Avenue, New York, NY 10036, United States, Barnard College, Columbia University, New York, NY, USA
² Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, United Kingdom
³ Departement Mathematik HG G 28, ETH, Rämistrasse 101, CH-8092 Zürich, Switzerland

Geometry & topology, Tome 21 (2017) no. 3, pp. 1731-1804.

Résumé

In the context of $CAT\left(0\right)$ cubical groups, we develop an analogue of the theory of curve complexes and subsurface projections. The role of the subsurfaces is played by a collection of convex subcomplexes called a factor system, and the role of the curve graph is played by the contact graph. There are a number of close parallels between the contact graph and the curve graph, including hyperbolicity, acylindricity of the action, the existence of hierarchy paths, and a Masur–Minsky-style distance formula.

We then define a hierarchically hyperbolic space; the class of such spaces includes a wide class of cubical groups (including all virtually compact special groups) as well as mapping class groups and Teichmüller space with any of the standard metrics. We deduce a number of results about these spaces, all of which are new for cubical or mapping class groups, and most of which are new for both. We show that the quasi-Lipschitz image from a ball in a nilpotent Lie group into a hierarchically hyperbolic space lies close to a product of hierarchy geodesics. We also prove a rank theorem for hierarchically hyperbolic spaces; this generalizes results of Behrstock and Minsky, of Eskin, Masur and Rafi, of Hamenstädt, and of Kleiner. We finally prove that each hierarchically hyperbolic group admits an acylindrical action on a hyperbolic space. This acylindricity result is new for cubical groups, in which case the hyperbolic space admitting the action is the contact graph; in the case of the mapping class group, this provides a new proof of a theorem of Bowditch.

Détail

DOI : 10.2140/gt.2017.21.1731

Classification : 20F36, 20F55, 20F65
Keywords: hierarchically hyperbolic, cube complexes, acylindrical, Teichmüller space, curve complex

Bibliographie

[1] I Agol, The virtual Haken conjecture, Doc. Math. 18 (2013) 1045

[2] H J Bandelt, V Chepoi, Metric graph theory and geometry: a survey, from: "Surveys on discrete and computational geometry" (editors J E Goodman, J Pach, R Pollack), Contemp. Math. 453, Amer. Math. Soc. (2008) 49 | DOI

[3] J A Behrstock, Asymptotic geometry of the mapping class group and Teichmüller space, Geom. Topol. 10 (2006) 1523 | DOI

[4] J Behrstock, R Charney, Divergence and quasimorphisms of right-angled Artin groups, Math. Ann. 352 (2012) 339 | DOI

[5] J Behrstock, M F Hagen, Cubulated groups: thickness, relative hyperbolicity, and simplicial boundaries, Groups Geom. Dyn. 10 (2016) 649 | DOI

[6] J Behrstock, M F Hagen, A Sisto, Hierarchically hyperbolic spaces II: Combination theorems and the distance formula, preprint (2015)

[7] J Behrstock, B Kleiner, Y Minsky, L Mosher, Geometry and rigidity of mapping class groups, Geom. Topol. 16 (2012) 781 | DOI

[8] J A Behrstock, Y N Minsky, Dimension and rank for mapping class groups, Ann. of Math. 167 (2008) 1055 | DOI

[9] N Bergeron, D T Wise, A boundary criterion for cubulation, Amer. J. Math. 134 (2012) 843 | DOI

[10] M Bestvina, K Bromberg, K Fujiwara, Bounded cohomology with coefficients in uniformly convex Banach spaces, preprint (2013)

[11] M Bestvina, K Bromberg, K Fujiwara, Constructing group actions on quasi-trees and applications to mapping class groups, Publ. Math. Inst. Hautes Études Sci. 122 (2015) 1 | DOI

[12] M Bestvina, M Feighn, Subfactor projections, J. Topol. 7 (2014) 771 | DOI

[13] M Bestvina, K Fujiwara, Bounded cohomology of subgroups of mapping class groups, Geom. Topol. 6 (2002) 69 | DOI

[14] M Bestvina, B Kleiner, M Sageev, Quasiflats in CAT(0) 2–complexes, Algebr. Geom. Topol. 16 (2016) 2663 | DOI

[15] B H Bowditch, Tight geodesics in the curve complex, Invent. Math. 171 (2008) 281 | DOI

[16] J F Brock, The Weil–Petersson metric and volumes of 3–dimensional hyperbolic convex cores, J. Amer. Math. Soc. 16 (2003) 495 | DOI

[17] J F Brock, R D Canary, Y N Minsky, The classification of Kleinian surface groups, II : The ending lamination conjecture, Ann. of Math. 176 (2012) 1 | DOI

[18] M Burger, N Monod, Continuous bounded cohomology and applications to rigidity theory, Geom. Funct. Anal. 12 (2002) 219 | DOI

[19] M Burger, S Mozes, Lattices in product of trees, Inst. Hautes Études Sci. Publ. Math. 92 (2000) 151

[20] P E Caprace, N Monod, Isometry groups of non-positively curved spaces: structure theory, J. Topol. 2 (2009) 661 | DOI

[21] P E Caprace, M Sageev, Rank rigidity for CAT(0) cube complexes, Geom. Funct. Anal. 21 (2011) 851 | DOI

[22] R Charney, An introduction to right-angled Artin groups, Geom. Dedicata 125 (2007) 141 | DOI

[23] R Charney, M W Davis, Finite K(π,1)s for Artin groups, from: "Prospects in topology" (editor F Quinn), Ann. of Math. Stud. 138, Princeton Univ. Press (1995) 110

[24] I Chatterji, G Niblo, From wall spaces to CAT(0) cube complexes, Internat. J. Algebra Comput. 15 (2005) 875 | DOI

[25] V Chepoi, Graphs of some CAT(0) complexes, Adv. in Appl. Math. 24 (2000) 125 | DOI

[26] V Chepoi, M F Hagen, On embeddings of CAT(0) cube complexes into products of trees via colouring their hyperplanes, J. Combin. Theory Ser. B 103 (2013) 428 | DOI

[27] M W Davis, The geometry and topology of Coxeter groups, 32, Princeton Univ. Press (2008)

[28] C Druţu, Quasi-isometry invariants and asymptotic cones, Internat. J. Algebra Comput. 12 (2002) 99 | DOI

[29] M G Durham, The augmented marking complex of a surface, J. Lond. Math. Soc. 94 (2016) 933 | DOI

[30] A Eskin, H Masur, K Rafi, Large scale rank of Teichmüller space, preprint (2013)

[31] K Fujiwara, Subgroups generated by two pseudo-Anosov elements in a mapping class group, I : Uniform exponential growth, from: "Groups of diffeomorphisms" (editors R Penner, D Kotschick, T Tsuboi, N Kawazumi, T Kitano, Y Mitsumatsu), Adv. Stud. Pure Math. 52, Math. Soc. Japan (2008) 283

[32] M F Hagen, The simplicial boundary of a CAT(0) cube complex, Algebr. Geom. Topol. 13 (2013) 1299 | DOI

[33] M F Hagen, Weak hyperbolicity of cube complexes and quasi-arboreal groups, J. Topol. 7 (2014) 385 | DOI

[34] M F Hagen, P Przytycki, Cocompactly cubulated graph manifolds, Israel J. Math. 207 (2015) 377 | DOI

[35] M F Hagen, D T Wise, Cubulating hyperbolic free-by-cyclic groups: the general case, Geom. Funct. Anal. 25 (2015) 134 | DOI

[36] F Haglund, Finite index subgroups of graph products, Geom. Dedicata 135 (2008) 167 | DOI

[37] F Haglund, D T Wise, Special cube complexes, Geom. Funct. Anal. 17 (2008) 1551 | DOI

[38] U Hamenstädt, Geometry of the mapping class groups, III: Quasi-isometric rigidity, preprint (2005)

[39] J Huang, Top dimensional quasiflats in CAT(0) cube complexes, preprint (2014)

[40] D Hume, Embedding mapping class groups into finite products of trees, preprint (2012)

[41] N V Ivanov, Automorphism of complexes of curves and of Teichmüller spaces, Internat. Math. Res. Notices (1997) 651 | DOI

[42] J Kahn, V Markovic, Immersing almost geodesic surfaces in a closed hyperbolic three manifold, Ann. of Math. 175 (2012) 1127 | DOI

[43] S H Kim, T Koberda, Embedability between right-angled Artin groups, Geom. Topol. 17 (2013) 493 | DOI

[44] S H Kim, T Koberda, The geometry of the curve graph of a right-angled Artin group, Internat. J. Algebra Comput. 24 (2014) 121 | DOI

[45] B Kleiner, The local structure of length spaces with curvature bounded above, Math. Z. 231 (1999) 409 | DOI

[46] J Maher, G Tiozzo, Random walks on weakly hyperbolic groups, preprint (2014)

[47] A I Mal’Cev, On a class of homogeneous spaces, Izvestiya Akad. Nauk. SSSR. Ser. Mat. 13 (1949) 9

[48] J F Manning, Geometry of pseudocharacters, Geom. Topol. 9 (2005) 1147 | DOI

[49] H A Masur, Y N Minsky, Geometry of the complex of curves, I : Hyperbolicity, Invent. Math. 138 (1999) 103 | DOI

[50] H A Masur, Y N Minsky, Geometry of the complex of curves, II : Hierarchical structure, Geom. Funct. Anal. 10 (2000) 902 | DOI

[51] P Mathieu, A Sisto, Deviation inequalities and CLT for random walks on acylindrically hyperbolic groups, preprint (2014)

[52] N Monod, Continuous bounded cohomology of locally compact groups, 1758, Springer (2001) | DOI

[53] A Nevo, M Sageev, The Poisson boundary of CAT(0) cube complex groups, Groups Geom. Dyn. 7 (2013) 653 | DOI

[54] G A Niblo, L D Reeves, Coxeter groups act on CAT(0) cube complexes, J. Group Theory 6 (2003) 399 | DOI

[55] D Osin, Acylindrically hyperbolic groups, Trans. Amer. Math. Soc. 368 (2016) 851 | DOI

[56] P Pansu, Croissance des boules et des géodésiques fermées dans les nilvariétés, Ergodic Theory Dynam. Systems 3 (1983) 415 | DOI

[57] P Pansu, Métriques de Carnot–Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. 129 (1989) 1 | DOI

[58] S D Pauls, The large scale geometry of nilpotent Lie groups, Comm. Anal. Geom. 9 (2001) 951 | DOI

[59] K Rafi, A combinatorial model for the Teichmüller metric, Geom. Funct. Anal. 17 (2007) 936 | DOI

[60] K Ruane, S Witzel, CAT(0) cubical complexes for graph products of finitely generated abelian groups, New York J. Math. 22 (2016) 637

[61] L Sabalka, D Savchuk, Submanifold projection, preprint (2012)

[62] M Sageev, Ends of group pairs and non-positively curved cube complexes, Proc. London Math. Soc. 71 (1995) 585 | DOI

[63] M Sageev, D T Wise, The Tits alternative for CAT(0) cubical complexes, Bull. London Math. Soc. 37 (2005) 706 | DOI

[64] P Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. 17 (1978) 555 | DOI

[65] A Sisto, Projections and relative hyperbolicity, Enseign. Math. 59 (2013) 165 | DOI

[66] S J Taylor, Right-angled Artin groups and Out(Fn), I : Quasi-isometric embeddings, Groups Geom. Dyn. 9 (2015) 275 | DOI

[67] W P Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. 19 (1988) 417 | DOI

[68] J H C Whitehead, Combinatorial homotopy, I, Bull. Amer. Math. Soc. 55 (1949) 213 | DOI

[69] J H C Whitehead, Combinatorial homotopy, II, Bull. Amer. Math. Soc. 55 (1949) 453 | DOI

[70] D T Wise, Cubulating small cancellation groups, Geom. Funct. Anal. 14 (2004) 150 | DOI

[71] D T Wise, Complete square complexes, Comment. Math. Helv. 82 (2007) 683 | DOI

[72] D T Wise, The structure of groups with a quasiconvex hierarchy, preprint (2011)

[73] D T Wise, From riches to raags : 3–manifolds, right-angled Artin groups, and cubical geometry, 117, Amer. Math. Soc. (2012) | DOI