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?

$\begingroup$ If you're interested in congruence arithmetic groups, consider: ams.org/mathscinetgetitem?mr=2251482 ams.org/mathscinetgetitem?mr=1654474 $\endgroup$ – Ian Agol Dec 20 '11 at 7:23
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 nonuniform (cusped) groups is the same. The question for maximal such groups is more interesting, and I am not sure what the answer is...

3$\begingroup$ 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