Classifying spaces, proper actions and stable homotopy theory

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Primary goal of this project is to compute invariants of discrete groups coming from stable homotopy theory. Secondary goal is to explore connections of these computations with number theory. The project deals with invariants of infinite discrete groups coming from stable homotopy theory, such as Morava K-theories and Morava E-theories. These invariants generalise the classical Euler characteristic and topological K-theory. For finite groups they are well-known and studied by Hopkins-Kuhn-Ravenel. For K-theory these invariants for infinite groups were computed by Lück and coathors. We plan to generalise both these computations.For Morava K-theories of infinite groups we expect to obtain a formula for the Morava K-theoretic Euler characteristic using centralisers. For Morava E-theory we plan to generalise the Hopkins-Kuhn-Ravenel character theory from finite to infinite groups. Finally, we will apply these results to concrete groups. Concrete formulas coming out of these should be of number theoretic nature. Some of them will contain values of Dedekind and Riemann zeta functions.