Cohomology of Bianchi Groups and Lefschetz Numbers

Bianchi groups are the congruence subgroups of SL(2,O) where O is the ring of integers of an imaginary quadratic field. In other words, they are "imaginary quadratic field" analogs of the classical modular groups, i.e. congruence subgroups of SL(2, Z). As in the classical case, the cohomology of Bianchi groups with certain coefficient modules are in fact the space of certain automorphic forms of cohomological type. Many problems about the cohomology of these arithmetic groups that are answered in their classical setting are still open. One of the fundamental open problem is to give a closed formula for the dimension of these cohomology spaces for a given Bianchi group (i.e. level) and coefficient system (i.e. weight). We will address this problem in the talk. After introducing Bianchi groups briefly, I will talk about a recent result on the dimension of the cohomology of Bianchi groups. More precisely, I will talk about a method, originally due to Harder, and how it is used to give a lower bound for the dimension of these cohomology spaces. This is a joint work with M. H. Sengun.