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Sep 11, 2021 at 9:26 comment added Radu T This might be too late, but here is a reference that might be of interest: Spectral Theory of Infinite-Area Hyperbolic Surfaces by Borthwick. The theory is indeed quite different but there are still many connections.
Dec 5, 2017 at 8:51 comment added Desiderius Severus I can also mention this related question and the reference quoted in the answer
Oct 30, 2017 at 1:25 comment added Kimball I don't know about things with infinite volume, but for non-arithmetic Fuchsian you don't get so many cusp forms. Sarnak has a survey which talks about this (maybe in the Bulletin), and Winnie Li has studied non-arithmetic groups. Also, in higher rank, all lattices are arithmetic by rigidity. For subgroups coming from quaternion algebras (which are arithmetic to me) there's the Jacquet-Langlands correspondence.
Oct 28, 2017 at 11:17 history edited Desiderius Severus
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Oct 28, 2017 at 6:55 history asked Desiderius Severus CC BY-SA 3.0