Introductory computations in the cohomology of arithmetic groups

Graham Ellis

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 Γ of SL2(Z), and then partially implements versions of the algorithms for the special linear group SL2(Od) over various rings of quadratic integers Od. The approach is also relevant for computations oncongruence subgroups of SL2(Od), m ≥ 2. 

Advanced Studies: Euro-Tbilisi Mathematical Journal, Special Issue (9 - 2021), pp. 1-31