Classifying spaces, proper actions and stable homotopy theory

本课题的主要目的是利用稳定同伦理论计算离散群的不变量。第二个目标是探索这些计算与数论的联系。本课题研究来自稳定同伦理论的无限离散群的不变量，如Morava k理论和Morava e理论。这些不变量推广了经典欧拉特征和拓扑k理论。对于有限群，它们是众所周知的，并被Hopkins-Kuhn-Ravenel研究过。对于k理论，无限群的这些不变量由l<s:1> ck和coathors计算。我们计划推广这两种计算。对于无限群的Morava k -理论，我们期望得到一个使用集中器的Morava k -理论欧拉特征的公式。对于Morava e理论，我们计划将Hopkins-Kuhn-Ravenel特征理论从有限群推广到无限群。最后，我们将这些结果应用到具体的群体中。从中得出的具体公式应该是数论性质的。其中一些会包含Dedekind和Riemann zeta函数的值。

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.