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
SL_{2}(Z), and then partially implements versions of the
algorithms for the special linear group SL_{2}(O_{d}) over
various rings of quadratic integers O_{d}. The approach is
also relevant for computations oncongruence subgroups of SL_{2}(O_{d}), m ≥ 2.
Advanced Studies: EuroTbilisi Mathematical Journal, Special Issue (9  2021), pp. 131
