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JMD

For nonuniform cofinite Fuchsian groups $\Gamma$ that satisfy a
certain additional geometric condition, we show that the Maass cusp
forms for $\Gamma$ are isomorphic to $1$-eigenfunctions of a
finite-term transfer operator. The isomorphism is constructive.

keywords:
geodesic flow.
,
symbolic dynamics
,
Maass cusp forms
,
period functions
,
transfer operator

DCDS

We construct cross sections for the geodesic flow on the orbifolds $\Gamma $\$ \mathbb{H}$ which are tailor-made for the requirements of transfer operator approaches to Maass cusp forms and Selberg zeta functions. Here, $\mathbb{H}$ denotes the hyperbolic plane and $\Gamma$ is a nonuniform geometrically finite Fuchsian group (not necessarily a lattice, not necessarily arithmetic) which satisfies an additional condition of geometric nature. The construction of the cross sections is uniform, geometric, explicit and algorithmic.

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