#### Volume 20, issue 2 (2020)

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The Segal conjecture for infinite discrete groups

### Wolfgang Lück

Algebraic & Geometric Topology 20 (2020) 965–986
##### 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.

##### Keywords
equivariant cohomotopy, Segal conjecture for infinite discrete groups
Primary: 55P91