We give a criterion in terms of the boundary for the existence of a proper cocompact action of a word-hyperbolic group on a CAT(0) cube complex. We describe applications towards lattices and hyperbolic 3-manifold groups. In particular, by combining the theory of special cube complexes, the surface subgroup result of Kahn-Markovic, and Agol’s criterion, we find that every subgroup separable clos...

In this paper we show that if a discrete group G acts properly isometrically on a discrete space X for which the uniform Roe algebra C * u (X) is exact then G is an exact group. As a corollary, we note that if the action is cocompact then the following are equivalent: The space X has Yu's property A; C * u (X) is exact; C * u (X) is nuclear.

We give another proof for a result of Brick ([2]) stating that the simple connectivity at infinity is a geometric property of finitely presented groups. This allows us to define the rate of vanishing of π∞ 1 for those groups which are simply connected at infinity. Further we show that this rate is linear for cocompact lattices in nilpotent and semi-simple Lie groups, and in particular for funda...

Abstract We prove sharp upper and lower bounds for generalized Calderón’s sums associated to frames on LCA groups generated by affine actions of cocompact subgroup translations and general measurable families of automorphisms. The proof makes use of techniques of analysis on metric spaces, and relies on a counting estimate of lattice points inside metric balls. We will deduce as special cases C...

We introduce a strong notion of quasiconvexity in finitely generated groups, which we call stability. Stability agrees with quasiconvexity in hyperbolic groups and is preserved under quasi-isometry for finitely generated groups. We show that the stable subgroups of mapping class groups are precisely the convex cocompact subgroups. This generalizes a well-known result of Behrstock [Beh06] and is...