MPEJ Volume 7, No. 1, 18pp. Received: May 31, 2001. Accepted: Sep 26, 2001. Mayer D., Neunhaeuserer J. 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 geodetic flow and the Eichler cohomology in number theory, see \cite{[CM3]}, \cite{[LW]}, \cite{[LZ]}. In \cite{[CM3]} it is indicated that such an isomorphism exists as well for the subgroups $\Gamma(2)$ and $\Gamma_{0}(2)$ of $PSL(2,\Z)$. We will prove this and provide some evidence by computer aided algebraic calculations that such an isomorphism exists for all principal congruence subgroups $\Gamma(N)$ and all congruence subgroups of Hecke type $\Gamma_{0}(N)$.