There are tons of material on the spectral theory of $L^2(\Gamma\backslash G)$ for a lattice $\Gamma$ in $G=PSL_2({\mathbb R})$. There are also many papers on the case of $\Gamma$ being convex-cocompact. But what about the intermediary case of Fuchsian groups of the first kind, which are not lattices?

Let me give an example: Let $\Lambda$ be a cocompact, torsion-free lattice in $G$. Then $\Lambda$ is the fundamental group of the Riemann surface $\Lambda\backslash G/K$ with $K=PSO(2)$. The structure of those is known: There are generators $\lambda_1,\dots,\lambda_{2g}$ with only one relation: $$ [\lambda_1,\lambda_2]\cdots [\lambda_{2g-1},\lambda_{2g}]=1. $$ So there exists a surjective group homomorphism $\chi:\Lambda\to\mathbb Z$. The kernel of $\chi$ would be the group I am interested in: $\Gamma=\ker(\chi)$.

So my question is: Is there an explicit description of the representation-theoretic spectrum of $L^2(\Gamma\backslash G)$? Is there a known theory of Eisenstein series which provides analytic continuation and functional equation?


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