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\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 \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.