Period functions for Maass wave forms and cohomology

The authors construct explicit isomorphisms between spaces of Maass wave forms and cohomology groups for discrete cofinite groups \Gamma\subset\mathrm{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 serie