Genus of arithmetic surface groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T03:30:33Z http://mathoverflow.net/feeds/question/82740 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/82740/genus-of-arithmetic-surface-groups Genus of arithmetic surface groups Katie 2011-12-05T21:38:00Z 2011-12-05T22:15:59Z <p>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? </p> http://mathoverflow.net/questions/82740/genus-of-arithmetic-surface-groups/82743#82743 Answer by Igor Rivin for Genus of arithmetic surface groups Igor Rivin 2011-12-05T22:15:59Z 2011-12-05T22:15:59Z <p>If there is an arithmetic group of genus \$2\$ (which there is, see <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa86/aa8626.pdf" rel="nofollow">http://matwbn.icm.edu.pl/ksiazki/aa/aa86/aa8626.pdf</a>), 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...</p>