Eisenstein series and the top degree cohomology of arithmetic subgroups of $SL_n/\mathbb{Q}$

Abstract

The cohomology H * (Γ, E) of a torsion-free arithmetic subgroup Γ of the special linear Q-group G = SLn may be interpreted in terms of the automorphic spectrum of Γ. Within this framework, there is a decomposition of the cohomology into the cuspidal cohomology and the Eisenstein cohomology The latter space is decomposed according to the classes {P} of associate proper parabolic Q-subgroups of G. Each summand H * {P} (Γ, E) is built up by Eisenstein series (or residues of such) attached to cuspidal automorphic forms on the Levi components of elements in {P}.The cohomology H * (Γ, E) vanishes above the degree given by the cohomological dimension cd(Γ) = n(n−1) 2. We are concerned with the internal structure of the cohomology in this top degree. On the one hand, we explicitly describe the associate classes {P} for which the corresponding summand H cd(Γ) {P} (Γ, E) vanishes. On the other hand, in the remaining cases of associate classes we construct various families of non-vanishing Eisenstein cohomology classes which span H cd(Γ) {Q} (Γ, C). Finally, in the case of a principal congruence subgroup Γ(q), q = p ν > 5, p ≥ 3 a prime, we give lower bounds for the size of these spaces if not even a precise formula for its dimension for certain associate classes {Q}.