28 nov 2023 -- 12:00
Euler characteristic is one of the basic invariants in algebraic topology. In this talk I will discuss various Euler characteristics for groups. The orbifold Euler characteristic of an infinite group has been well studied and is related to number theory and geometry. In particular, by the results of Harder, Harer-Zagier, Brown and others, for certain arithmetic groups and mapping class groups, the values of the orbifold Euler characteristic can be described in terms of special values of zeta functions. On the other hand, for finite groups a well-studied invariant is Morava K-theoretic Euler characteristic. This is an invariant coming from chromatic homotopy theory and there is an explicit formula due to Hopkins-Kuhn-Ravenel. I’ll provide a new formula for Morava K-theoretic Euler characteristic of an infinite group satisfying some finiteness properties and discuss its relations with the orbifold Euler characteristic and number theory. In particular I will present explicit computations of the Morava K-theoretic Euler characteristic of some arithmetic groups in terms of class numbers and special values of zeta functions.