# EZ-structures and topological applications

### F. Thomas Farrell

S.U.N.Y. Binghamton, USA### Jean-François Lafont

S.U.N.Y. Binghamton, USA

## 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 $δ$-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 $(D_{n},Δ)$, where $n$ is sufficiently large, and $Δ$ is a closed subset of $∂D_{n}=S_{n−1}$. The action has the property that it is proper and cocompact on $D_{n}−Δ$, and that if $K⊂D_{n}−Δ$ is compact, that $diam(gK)$ tends to zero as $g→∞$. We call this property $(∗_{Δ})$.

Our second theorem uses techniques of Farrell–Hsiang [8] to show that the Novikov conjecture holds for any torsion-free discrete group satisfying condition $(∗_{Δ})$ (giving a new proof that torsion-free $δ$-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 $δ$-hyperbolic group $Γ$, we can obtain a lower bound for the homotopy groups $π_{n}(P(BΓ))$, where $P(⋅)$ is the stable topological pseudo-isotopy functor.

## Cite this article

F. Thomas Farrell, Jean-François Lafont, EZ-structures and topological applications. Comment. Math. Helv. 80 (2005), no. 1, pp. 103–121

DOI 10.4171/CMH/7