Almost flat K-theory of classifying spaces

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 $\mathrm{\Gamma }$ with finite classifying space $B\mathrm{\Gamma }$, we study a correspondence between between almost flat K-theory classes on $B\mathrm{\Gamma }$ and group homomorphism $K_0(C^{\ast }(\mathrm{\Gamma }))\to \mathbb{Z}$ that are implemented by pairs of discrete asymptotic homomorphisms from $C^{\ast }(\mathrm{\Gamma })$ to matrix algebras.