An improved method for computing group cohomology of congruence subgroups of SL3(Z)

A well-known theorem due to Manin [16] gives a relationship between modular symbols for a congruence subgroup Gamma0( N) of SL2( Z ), and the homology of X0(N). A corresponding theorem for congruence subgroups of SL3( Z ) was made by Avner Ash [3]. In this paper, we demonstrate an improved method for computing the group cohomology for congruence subgroups of SL3( Z ). For W a Gamma0(N)-module, we identify the group cohomology of Gamma0(N) with a subspace of Wa, for some integer a. This method uses a generalized notion of Grobner bases in order to determine a minimal generating set for the ideal of conditions describing the desired subspace of Wa.