Accessing the cohomology of discrete groups above their virtual cohomological dimension

TL;DR

This paper presents a new method to explicitly compute the Farrell-Tate cohomology of discrete groups, applying it to Bianchi groups and related groups, revealing that their cohomology is determined by finite subgroup conjugacy classes.

Contribution

The paper introduces a novel approach to determine Farrell-Tate cohomology directly from group structure, independent of geometric models, and provides explicit formulas for Bianchi groups.

Findings

Farrell-Tate cohomology of Bianchi groups is determined by conjugacy classes of finite subgroups.

New formulas for counting conjugacy classes in Bianchi groups are derived and numerically evaluated.

The cohomology ring structure of Bianchi groups is characterized using homological torsion insights.

Abstract

We introduce a method to explicitly determine the Farrell-Tate cohomology of discrete groups. We apply this method to the Coxeter triangle and tetrahedral groups as well as to the Bianchi groups, i.e. PSL_2 over the ring of integers in an imaginary quadratic number field, and to their finite index subgroups. We show that the Farrell-Tate cohomology of the Bianchi groups is completely determined by the numbers of conjugacy classes of finite subgroups. In fact, our access to Farrell-Tate cohomology allows us to detach the information about it from geometric models for the Bianchi groups and to express it only with the group structure. Formulae for the numbers of conjugacy classes of finite subgroups in the Bianchi groups have been determined in a thesis of Kr\"amer, in terms of elementary number-theoretic information on the ring of integers. An evaluation of these formulae for a large number of Bianchi groups is provided numerically in the appendix. Our new insights about the homological torsion allow us to give a conceptual description of the cohomology ring structure of the Bianchi groups.