Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity

By Jason A. Behrstock, Cornelia Drutu, Lee Mosher. 2005

191 citations recorded by the source corpus.

We study the geometry of non-relatively 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 non-relatively 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 non-hyperbolic relative to any non-trivial 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. Non-uniform lattices in higher rank semisimple Lie groups are thick and hence non-relatively hyperbolic, in contrast with rank one which provided the motivating examples of relatively hyperbolic groups. Mapping class groups are the first examples of non-relatively 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fc823cf15ed2df02382a1ad085a9961ed0bacd02

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Citations in the corpus

Statistically convex-cocompact actions of groups with contracting elements (2016, background): “received much attention. Indeed, one of the motivations behind the present study is the application of the approach presented here to Mod. Most mapping class groups are not relatively hyperbolic, cf. [9]. Nevertheless, their action on Teichmul ler spaces exhibits very interesting negative behavior, ”
Geometry and rigidity of mapping class groups (2008, background): “We note that by results of [4] such pieces can not be realized as asymptotic cones of subgroups of M.”
Relatively hyperbolic groups: geometry and quasi-isometric invariance (2006, background): “ted, that is they have at least one asymptotic cone without (global) cut-points. Under this assumption it can be ensured that each H′ i is quasi-isometric to some Hj. The theorem has been improved in [BDM05], where it is only required that each Hi is not properly relatively hyperbolic. This type of ”
Groups acting on tree-graded spaces and splittings of relatively hyperbolic groups☆ (2006, background): “genus of the corresponding surface plus the number of punctures is at least 6 [B]; • Fundamental groups of graph-manifolds which are not Sol nor Nil manifolds [KKL]; • Many right angled Artin groups ([BDM]). Note that mapping class groups, Teichmu¨ller spaces and fundamental groups of graphmanifolds”
Divergence and quasimorphisms of right-angled Artin groups (2010, background): “sic in XΓ has divergence which is either linear or at least quadratic. Theorem 5.2 states that AΓ contains a rank-one geodesic, hence the divergence of AΓ is at least quadratic. On the other hand, in [BDM] it is shown that for any connected graph Γ, AΓ is algebraicallythick of order at most 1. It is”
The geometry of the disk complex (2010, background): “Examples in the literature include the marking graph [31], the pants graph [11, 2], the Hatcher-Thurston graph [23], the complex of separating curves [9], the arc complex and the curve complex.”