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It is known that for each genus, only finitely many points in the moduli space of hyperbolic genus g surfaces are arithmetic. I'm wondering if an existence result is known: for which g do we have examples of arithmetic Fuchsian genus g surface groups? What about surfaces with boundary?

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If there is an arithmetic group of genus $2$ (which there is, see http://matwbn.icm.edu.pl/ksiazki/aa/aa86/aa8626.pdf), then there are such of all genera, by taking finite index subgroups. The argument for non-uniform (cusped) groups is the same. The question for maximal such groups is more interesting, and I am not sure what the answer is...

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    $\begingroup$ One could also ask for congruence groups, which is more general than maximal arithmetic groups but still much more restrictive than arbitrary arithmetic groups, perhaps sufficiently so to be "interesting". $\endgroup$ – Noam D. Elkies Dec 6 '11 at 5:16

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