Almost flat K-theory of classifying spaces

Abstract

We give a rigorous account and prove continuity properties for the correspondence between almost flat bundles on a triangularizable compact connected space and the quasi-representations of its fundamental group. For a discrete countable group \Gamma with finite classifying space B\Gamma , we study a correspondence between almost flat K-theory classes on B\Gamma and group homomorphism K_0(C^*(\Gamma)) \to \mathbb{Z} that are implemented by pairs of discrete asymptotic homomorphisms from C^*(\Gamma) to matrix algebras.