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The Segal conjecture for infinite discrete groups

Wolfgang Lück

Algebraic & Geometric Topology 20 (2020) 965–986

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 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 E¯G for I the augmentation ideal.

equivariant cohomotopy, Segal conjecture for infinite discrete groups
Mathematical Subject Classification 2010
Primary: 55P91
Received: 26 January 2019
Revised: 5 July 2019
Accepted: 9 August 2019
Published: 23 April 2020
Wolfgang Lück
Mathematisches Institut
Rheinische Wilhelms-Universität Bonn