Equivariant vector bundles over classifying spaces for proper actions

Abstract

Let $G$ be an infinite discrete group and let $\underline{E}G$ be a classifying space for proper actions of $G$. Every $G$-equivariant vector bundle over $\underline{E}G$ gives rise to a compatible collection of representations of the finite subgroups of $G$. We give the first examples of groups $G$ with a cocompact classifying space for proper actions $\underline{E}G$ admitting a compatible collection of representations of the finite subgroups of $G$ that does not come from a $G$-equivariant (virtual) vector bundle over $\underline{E}G$. This implies that the Atiyah-Hirzeburch spectral sequence computing the $G$-equivariant topological $K$-theory of $\underline{E}G$ has non-zero differentials. On the other hand, we show that for right angled Coxeter groups this spectral sequence always collapes at the second page and compute the $K$-theory of the classifying space of a right angled Coxeter group.