The automorphic cohomology group attached to an arithmetic subgroup of a semi-simple algebraic group G, defined over the rationals and of positive rank, vanishes above the degree given by the cohomological dimension. In this talk, in terms of Eisenstein series, we are concerned with the internal structure of the cohomology in this top degree. In the case of congruence subgroups of the special linear group, the actual construction of non-vanishing Eisenstein cohomology classes allows to give lower bounds for the dimension of this space.