Group quasi-representations and almost flat bundles

Abstract

We study the existence of quasi-representations of discrete groups $G$ into unitary groups $U(n)$ that induce prescribed partial maps $K_0(C^*(G))\to \mathbb{Z}$ on the K-theory of the group C*-algebra of $G$. We give conditions for a discrete group $G$ under which the K-theory group of the classifying space $BG$ consists entirely of almost flat classes.