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?
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...