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 is finite is replaced in our setting by the assumption that there exists a finite model for the classifying space 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 that the zeroth stable cohomotopy of the classifying space is isomorphic to the –adic completion of the ring given by the zeroth equivariant stable cohomotopy of for the augmentation ideal.
"The Segal conjecture for infinite discrete groups." Algebr. Geom. Topol. 20 (2) 965 - 986, 2020. https://doi.org/10.2140/agt.2020.20.965