Asymptotic Lipschitz maps, combable groups and higher signatures (Q1976289)

Let \(\Gamma\) be a torsion-free finitely generated group admitting a proper combing of bounded multiplicity. It is proved that in this case the analytic assembly map is rationally injective. In particular, the higher signatures of \(\Gamma\) are oriented homotopy invariants. The proof is based on a family of asymptotically proper Lipschitz maps from Rips complexes to Euclidean spaces used to construct a natural morphism \(E(\widetilde{P}_i/\Gamma,*)\to E(C^*(\Gamma),{\mathbb C})\), where \(\widetilde{P}_i/\Gamma\) are tubular neighbourhoods of a family of approximations of \(B\Gamma\) by finite complexes and \(E\) denotes \(E\)-theory of Connes and Higson.