In the context of CAT(0) 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-Minsky, Eskin-Masur-Rafi, Hamenstädt, and 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.

We introduce a number of tools for finding and studying hierarchically hyperbolic spaces (HHS), a rich class of spaces including mapping class groups of surfaces, Teichmüller space with either the Teichmüller or Weil-Petersson metrics, right-angled Artin groups, and the universal cover of any compact special cube complex. We begin by introducing a streamlined set of axioms defining an HHS. We prove that all HHS satisfy a Masur-Minsky-style distance formula, thereby obtaining a new proof of the distance formula in the mapping class group without relying on the Masur-Minsky hierarchy machinery. We then study examples of HHS; for instance, we prove that when M is a closed irreducible 3-manifold then π1M is an HHS if and only if it is neither N il nor Sol. We establish this by proving a general combination theorem for trees of HHS (and graphs of HH groups). We also introduce a notion of "hierarchical quasiconvexity", which in the study of HHS is analogous to the role played by quasiconvexity in the study of Gromov-hyperbolic spaces.

Abstract. We study the geometry of nonrelatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup. This implies that a group being relatively hyperbolic with nonrelatively hyperbolic peripheral subgroups is a quasi-isometry invariant. As an application, Artin groups are relatively hyperbolic if and only if freely decomposable.We also introduce a new quasi-isometry invariant of metric spaces called metrically thick, which is sufficient for a metric space to be nonhyperbolic relative to any nontrivial collection of subsets. Thick finitely generated groups include: mapping class groups of most surfaces; outer automorphism groups of most free groups; certain Artin groups; and others. Nonuniform lattices in higher rank semisimple Lie groups are thick and hence nonrelatively hyperbolic, in contrast with rank one which provided the motivating examples of relatively hyperbolic groups. Mapping class groups are the first examples of nonrelatively hyperbolic groups having cut points in any asymptotic cone, resolving several questions of Drutu and Sapir about the structure of relatively hyperbolic groups. Outside of group theory, Teichmüller spaces for surfaces of sufficiently large complexity are thick with respect to the Weil-Peterson metric, in contrast with Brock-Farb's hyperbolicity result in low complexity.

In this work, we study the asymptotic geometry of the mapping class group and Teichmuller space. We introduce tools for analyzing the geometry of "projection" maps from these spaces to curve complexes of subsurfaces; from this we obtain information concerning the topology of their asymptotic cones. We deduce several applications of this analysis. One of which is that the asymptotic cone of the mapping class group of any surface is tree-graded in the sense of Druţu and Sapir; this treegrading has several consequences including answering a question of Druţu and Sapir concerning relatively hyperbolic groups. Another application is a generalization of the result of Brock and Farb that for low complexity surfaces Teichmüller space, with the Weil-Petersson metric, is ı -hyperbolic. Although for higher complexity surfaces these spaces are not ı -hyperbolic, we establish the presence of previously unknown negative curvature phenomena in the mapping class group and Teichmüller space for arbitrary surfaces.20F67; 30F60, 57M07, 20F65

We study the large scale geometry of mapping class groups MCG.S/, using hyperbolicity properties of curve complexes. We show that any self quasi-isometry of MCG.S / (outside a few sporadic cases) is a bounded distance away from a leftmultiplication, and as a consequence obtain quasi-isometric rigidity for MCG.S /, namely that groups quasi-isometric to MCG.S / are equivalent to it up to extraction of finite-index subgroups and quotients with finite kernel. (The latter theorem was proved by Hamenstädt using different methods).As part of our approach we obtain several other structural results: a description of the tree-graded structure on the asymptotic cone of MCG.S/; a characterization of the image of the curve complex projections map from MCG.S/ to Q Y ÂS C.Y /; and a construction of †-hulls in MCG.S /, an analogue of convex hulls.

In this paper we explore relationships between divergence and thick groups, and with the same techniques we estimate lengths of shortest conjugators. We produce examples, for every positive integer n, of CAT (0) groups which are thick of order n and with polynomial divergence of order n + 1, both these phenomena are new. With respect to thickness, these examples show the non-triviality at each level of the thickness hierarchy defined by Behrstock-Druţu-Mosher in [BDM09]. With respect to divergence our examples resolve questions of Gromov [Gro93] and Gersten [Ger94a] [Ger94b] (the divergence questions were also recently and independently answered by Macura [Mac]). We also provide general tools for obtaining both lower and upper bounds on the divergence of geodesics and spaces, and we give the definitive lower bound for Morse geodesics in the CAT (0) spaces, generalizing earlier results of Kapovich-Leeb [KL98] and Bestvina-Fujiwara [BF]. In the final section, we turn to the question of bounding the length of the shortest conjugators in several interesting classes of groups. We obtain linear and quadratic bounds on such lengths for classes of groups including 3-manifold groups and mapping class groups (the latter gives new proofs of corresponding results of Masur-Minsky in the pseudo-Anosov case [MM00] and Tao in the reducible case [Tao11]).

Abstract. We give a group theoretic characterization of geodesics with superlinear divergence in the Cayley graph of a right-angled Artin group A Γ with connected defining graph. We use this to determine when two points in an asymptotic cone of A Γ are separated by a cut-point. As an application, we show that if Γ does not decompose as the join of two subgraphs, then A Γ has an infinite-dimensional space of non-trivial quasimorphisms. By the work of Burger and Monod, this leads to a superrigidity theorem for homomorphisms from lattices into right-angled Artin groups.

We show that the fundamental groups of any two closed irreducible non-geometric graph-manifolds are quasi-isometric. This answers a question of Kapovich and Leeb. We also classify the quasi-isometry types of fundamental groups of graph-manifolds with boundary in terms of certain finite two-colored graphs. A corollary is the quasi-isometry classification of Artin groups whose presentation graphs are trees. In particular any two right-angled Artin groups whose presentation graphs are trees of diameter at least 3 are quasi-isometric, answering a question of Bestvina; further, this quasi-isometry class does not include any other right-angled Artin groups.Comment: Revised on referee's comments to add more details in proof. Introduction revised for clarit