EZ-structures and topological applications

Abstract

In this paper, we introduce the notion of an EZ-structure on a group, an equivariant version of the Z-structures introduced by Bestvina [4]. Examples of groups having an EZ-structure include (1) torsion free $\delta$-hyperbolic groups, and (2) torsion free $\mathrm{CAT}(0)$-groups.

Our first theorem shows that any group having an EZ-structure has an action by homeomorphisms on some $({\mathbb{D}}^n,\Delta )$, where $n$ is sufficiently large, and $\Delta$ is a closed subset of $\partial {\mathbb{D}}^n=S^{n-1}$. The action has the property that it is proper and cocompact on ${\mathbb{D}}^n-\Delta$, and that if $K\subset {\mathbb{D}}^n-\Delta$ is compact, that $\mathrm{diam}(gK)$ tends to zero as $g\to \infty$. We call this property $({\ast }_{\Delta })$.

Our second theorem uses techniques of Farrell–Hsiang [8] to show that the Novikov conjecture holds for any torsion-free discrete group satisfying condition $({\ast }_{\Delta })$ (giving a new proof that torsion-free $\delta$-hyperbolic and $\mathrm{CAT}(0)$ groups satisfy the Novikov conjecture).

Our third theorem gives another application of our main result. We show how, in the case of a torsion-free $\delta$-hyperbolic group $\Gamma$, we can obtain a lower bound for the homotopy groups ${\pi }_n(\mathcal{P}(B\Gamma ))$, where $\mathcal{P}(\cdot )$ is the stable topological pseudo-isotopy functor.