An isomorphism between polynomial eigenfunctions of the transfer operator and the Eichler cohomology for modular groups

Abstract

For the group PSL(2,Z) it is known that there is an isomorphism between polynomial eigenfunctions of the transfer operator for the geodesic flow and the Eichler cohomology in the theory of modular forms. In a recent paper by Chang and Mayer it is indicated that such an isomorphism exists as well for certain subgroups of PSL(2,Z). We prove this and provide some evidence by computer aided algebraic calculations that such an isomorphism exists for all principal congruence subgroups and all congruence subgroups of Hecke type.