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.
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