  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.

Algebraic Topology;Group Theory;K-Theory and Homology