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

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