  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 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.
