On the Dimension of Cohomology of Bianchi Groups

TL;DR

This paper establishes lower bounds for the cuspidal cohomology of Bianchi groups using Lefschetz numbers, enhancing understanding of their cohomological properties and relating involutions to base change classes.

Contribution

It introduces a novel method using Lefschetz numbers to estimate cohomology dimensions of Bianchi groups, complementing existing asymptotic results.

Findings

Provides explicit lower bounds for cuspidal cohomology

Connects involutions with base change classes in cohomology

Complements recent asymptotic cohomology results

Abstract

Using Lefschetz numbers of certain involutions, we provide lower bounds for the cuspidal cohomology of principal congruence subgroups of Bianchi groups. The asymptotic lower bounds that follow from our results complement recent results of Calegari-Emerton, Marshall and Finis-Grunewald-Tirao. Moreover, we discuss the relationship between these involutions and the base change classes in the cohomology.