Eichler-Shimura homology and the finite generation of cusp forms by hyperbolic Poincaré series
Out of a quite general theorem concerning the span of decomposable classes in the first homology group of a finitely generated group with coefficients in an irreducible representation the authors obtain some new results on the finite generation of cusp forms by hyperbolic Poincaré series with respect to a discrete subgroup of \(SL_ 2({\mathbb{R}})\). Given a finitely generated group \(\Gamma\) and an absolutely irreducible nontrivial direct summand \(\sigma\) of \(Adjoint. \circ \rho\) where \(\rho\) is an absolutely irreducible finite dimensional representation of the underlying Lie group then it is proved that \(H_ 1(\Gamma,\sigma^*)\) is spanned by decomposable classes. Applying this result in the situation \(\Gamma \subset SL_ 2({\mathbb{R}})\) and \(\rho\) the m-th symmetric power of the standard representation and using Eichler-Shimura theory the authors describe a way how to pass from a set of generators for \(\Gamma\) to a finite spanning set for the space \(S_{2m}(\Gamma)\) of cusp forms with respect to \(\Gamma\).
scientific article; zbMATH DE number 4002180