The authors construct explicit isomorphisms between spaces of Maass wave forms and cohomology groups for discrete cofinite groups $Gammasubsetmathrm{PSL}_2({mathbb{R}})$.

In the case that $Gamma$ is the modular group $mathrm{PSL}_2({mathbb{Z}})$ this gives a cohomological framework for the results in Period functions for Maass wave forms. I, of J. Lewis and D. Zagier in Ann. Math. 153 (2001), 191-258, where a bijection was given between cuspidal Maass forms and period functions.

The authors introduce the concepts of mixed parabolic cohomology group and semi-analytic vectors in principal series representation. This enables them to describe cohomology groups isomorphic to spaces of Maass cusp forms, spaces spanned by residues of Eisenstein series, and spaces of all $Gamma$-invariant eigenfunctions of the Laplace operator.

For spaces of Maass cusp forms the authors also describe isomorphisms to parabolic cohomology groups with smooth coefficients and standard cohomology groups with distribution coefficients. They use the latter correspondence to relate the Petersson scalar product to the cup product in cohomology.