K-THEORY AND ACTIONS ON EUCLIDEAN RETRACTS

Abstract

This note surveys axiomatic results for the Farrell-Jones Conjecture in terms of actions on Euclidean retracts and applications of these to GLn(Z), relative hyperbolic groups and mapping class groups. IntroductionMotivated by surgery theory Hsiang [43] made a number of influential conjectures about the K-theory of integral group rings Z[G] for torsion free groups G. These conjectures often have direct implications for the classification theory of manifolds of dimension ≥ 5. A good example is the following. An h-cobordism is a compact manifold W that has two boundary components M 0 and M 1 such that both inclusions M i → W are homotopy equivalences. The Whitehead group Wh(G) is the quotient of K 1 (Z[G]) by the subgroup generated by the canonical units ±g, g ∈ G. Associated to an h-cobordism is an invariant, the Whitehead torsion, in Wh(G), where G is the fundamental group of W . A consequence of the s-cobordism theorem is that for dim W ≥ 6, an h-cobordism W is trivial (i.e., isomorphic to a product M 0 ×[0, 1]) iff its Whitehead torsion vanishes. Hsiang conjectured that for G torsion free Wh(G) = 0, and thus that in many cases h-cobordisms are products.The Borel conjecture asserts that closed aspherical manifolds are topologically rigid, i.e., that any homotopy equivalence to another closed manifold is homotopic to a homeomorphism. The last step in proofs of instances of this conjecture via surgery theory uses a vanishing result for Wh(G) to conclude that an h-cobordism is a product and that therefore the two boundary components are homeomorphic.Farrell-Jones [28] pioneered a method of using the geodesic flow on non-positively curved manifolds to study these conjectures. This created a beautiful connection between K-theory and dynamics that led Farrell-Jones [30], among many other results, to a proof of the Borel Conjecture for closed Riemannian manifolds of nonpositive curvature of dimension ≥ 5. Moreover, Farrell-Jones [29] formulated (and proved many instances of) a conjecture about the structure of the algebraic Ktheory (and L-theory) of group rings, even in the presence of torsion in the group. Roughly, the Farrell-Jones Conjecture states that the main building blocks for the K-theory of Z[G] is the K-theory of Z[V ] where V varies of the family of virtually cyclic subgroups of G. It implies a number of other conjectures, among them Hsiang's conjectures, the Borel Conjecture in dimension ≥ 5, the Novikov Conjecture on the homotopy invariant of higher signatures, Kaplansky's conjecture about idempotents in group rings, see [54] for a summary of these and other applications.My goal in this note is twofold. The first goal is to explain a condition formulated in terms of existence of certain actions of G on Euclidean retracts that implies the Farrell-Jones Conjecture for G. This condition was developed in joint work with Lück and Reich [8,12] where the connection between K-theory and dynamics has