EZ-structures and topological applications

Tom Farrell, Jean-François Lafont
2005 Commentarii Mathematici Helvetici  
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 , ∆), where n is sufficiently large, and ∆ is a closed subset of ∂D n = S n−1 . The action has the property that it is proper
more » ... that it is proper and cocompact on D n − ∆, and that if K ⊂ D n − ∆ is compact, that diam(gK) tends to zero as g → ∞. We call this property ( * ∆ ). Our second theorem uses techniques of Farrell-Hsiang [8] to show that the Novikov conjecture holds for any torsion-free discrete group satisfying condition ( * ∆ ) (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.
doi:10.4171/cmh/7 fatcat:6zgpsexl7rgazmkuhzw5fe44wi