MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
If you're interested in congruence arithmetic groups, consider: – Ian Agol Dec 20 '11 at 7:23

If there is an arithmetic group of genus $2$ (which there is, see, 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...

share|cite|improve this answer
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". – Noam D. Elkies Dec 6 '11 at 5:16

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.