The mod 2 cohomology rings of congruence subgroups in the Bianchi groups

TL;DR

This paper introduces new computational tools and methods for analyzing torsion in the cohomology of congruence subgroups within Bianchi groups, focusing on fundamental domains and spectral sequences.

Contribution

It presents an algorithm for fundamental domain computation and analyzes spectral sequences to facilitate torsion calculations in Bianchi groups' cohomology.

Findings

New algorithm for fundamental domain calculation

Analysis of equivariant spectral sequence

Reduction techniques for torsion subcomplexes

Abstract

We provide new tools for the calculation of the torsion in the cohomology of congruence subgroups in the Bianchi groups : An algorithm for finding particularly useful fundamental domains, and an analysis of the equivariant spectral sequence combined with torsion subcomplex reduction.\_\_\_\_\_\_\_\_\_\_With an appendix by Bui Anh Tuan and Sebastian Sch{\"o}nnenbeck