JournalscmhVol. 80 , No. 1DOI 10.4171/cmh/7

EZ-structures and topological applications

  • F. Thomas Farrell

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

    S.U.N.Y. Binghamton, USA
EZ-structures and topological applications cover

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\delta-hyperbolic groups, and (2) torsion free CAT(0)CAT(0)-groups. Our first theorem shows that any group having an EZ-structure has an action by homeomorphisms on some (\mDn,Δ)(\mD^n, \Delta), where nn is sufficiently large, and Δ\Delta is a closed subset of \mDn=Sn1\partial \mD^n=S^{n-1}. The action has the property that it is proper and cocompact on \mDnΔ\mD^n-\Delta, and that if K\mDnΔK\subset \mD^n-\Delta is compact, that diam(gK)diam(gK) tends to zero as gg\rightarrow \infty. We call this property (Δ)(*_\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 (Δ)(*_\Delta) (giving a new proof that torsion-free δ\delta-hyperbolic and CAT(0)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\delta-hyperbolic group Γ\Gamma, we can obtain a lower bound for the homotopy groups πn(P(BΓ))\pi_n(\mathcal P(B\Gamma)), where P()\mathcal P(\cdot ) is the stable topological pseudo-isotopy functor.