The Segal conjecture for infinite discrete groups

Abstract

We formulate and prove a version of the Segal conjecture for infinite groups. For finite groups it reduces to the original version. The condition that $G$ is finite is replaced in our setting by the assumption that there exists a finite model for the classifying space $\underset{¯}{E}G$ for proper actions. This assumption is satisfied for instance for word hyperbolic groups or cocompact discrete subgroups of Lie groups with finitely many path components. As a consequence we get for such groups $G$ that the zeroth stable cohomotopy of the classifying space $BG$ is isomorphic to the $I$–adic completion of the ring given by the zeroth equivariant stable cohomotopy of $\underset{¯}{E}G$ for $I$ the augmentation ideal.