Introductory computations in the cohomology of arithmetic groups

TL;DR

This paper introduces computer algorithms for calculating the integral cohomology of arithmetic groups, emphasizing homotopy and perturbation methods over cellular subdivision, with applications to Hecke operators on congruence subgroups.

Contribution

It presents novel algorithms for computing integral cohomology and Hecke operators on arithmetic groups, extending to groups over quadratic integer rings and higher dimensions.

Findings

Algorithms successfully compute Hecke operators on integral cohomology.

Implementation covers congruence subgroups of SL_2(Z) and SL_2(O_d).

Approach improves computational methods by avoiding cellular subdivision.

Abstract

This paper describes an approach to computer aided calculations in the cohomology of arithmetic groups. It complements existing literature on the topic by emphasizing homotopies and perturbation techniques, rather than cellular subdivision, as the tools for implementing on a computer topological constructions that fail to preserve cellular structures Furthermore, it focuses on calculating integral cohomology rather than just rational cohomology or cohomology at large primes. In particular, the paper describes and fully implements algorithms for computing Hecke operators on the integral cuspidal cohomology of congruence subgroups $\Gamma$ of $SL_2(\mathbb Z)$, and then partially implements versions of the algorithms for the special linear group $SL_2({\cal O}_d)$ over various rings of quadratic integers ${\cal O}_d$. The approach is also relevant for computations on congruence subgroups of $SL_m({\cal O}_d)$, $m\ge 2$.