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The Phillips-Sarnak conjecture states that for a generic Fuchsian lattice the space of Maass cusp forms is finite-dimensional. Generic here means in particular non-uniform, non-arithmetic, no special symmetries.

When it comes to higher dimensional hyperbolic spaces then the isometry group still contains non-uniform, non-arithmetic lattices. Therefore I wonder what is known or expected about the existence or non-existence of Maass cusp forms for these lattices. For example, is there is an analogon of the Phillips-Sarnak conjecture?

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  • $\begingroup$ What about superrigidity? Does it not contradict your claim about existence of non-arithmetic lattices in higher rank? en.wikipedia.org/wiki/Superrigidity $\endgroup$
    – Marc Palm
    Commented Apr 17, 2014 at 8:31
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    $\begingroup$ Hyperbolic spaces, also higher dimensional ones, are rank one spaces. $\endgroup$
    – Maik Köster
    Commented Apr 22, 2014 at 6:20

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