On the Integral Cohomology of Bianchi groups

TL;DR

This paper presents extensive computational analysis of the integral cohomology of Euclidean Bianchi groups and their subgroups, revealing patterns in torsion growth and properties relevant to Hecke eigenvalues.

Contribution

It provides the first large-scale systematic computational data on the integral cohomology of Bianchi groups, including insights into torsion asymptotics and liftibility of Hecke eigenvalues.

Findings

Data on torsion growth in homology

Insights into liftibility of Hecke eigenvalues

Patterns in cohomology of Bianchi groups

Abstract

Extensive and systematic machine computations are carried out to investigate the integral cohomology of the Euclidean Bianchi groups and their congruence subgroups. The collected data give insight on several aspects, including the asymptotic behaviour of the torsion in the first homology. Along with the experimental work, some basic properties of the integral cohomology are recorded with an eye towards the liftibility issue of Hecke eigenvalue systems.