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 $CAT(0)$-groups. Our first theorem shows that any group having an EZ-structure has an action by homeomorphisms on some $(\mD^n, \Delta)$, where $n$ is sufficiently large, and $\Delta$ is a closed subset of $\partial \mD^n=S^{n-1}$. The action has the property that it is proper and cocompact on $\mD^n-\Delta$, and that if $K\subset \mD^n-\Delta$ is compact, that $diam(gK)$ tends to zero as $g\rightarrow \infty$. We call this property $(*_\Delta)$. Our second theorem uses techniques of Farrell-Hsiang [8] to show that the Novikov conjecture holds for any torsion-free discrete group satisfying condition $(*_\Delta)$ (giving a new proof that torsion-free $\delta$-hyperbolic and $CAT(0)$ 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.