Arithmetic Fuchsian group - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T05:32:30Z http://mathoverflow.net/feeds/question/78553 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/78553/arithmetic-fuchsian-group Arithmetic Fuchsian group Karl 2011-10-19T09:08:05Z 2011-10-19T09:29:03Z <p>Dear all,</p> <p>I have the following questions: Are all Fuchsian groups of signature $(0;2,2,2,\infty)$ arithmetic? What is known about the trace fields of these groups?</p> <p>Best, K. </p> http://mathoverflow.net/questions/78553/arithmetic-fuchsian-group/78555#78555 Answer by Igor Rivin for Arithmetic Fuchsian group Igor Rivin 2011-10-19T09:29:03Z 2011-10-19T09:29:03Z <p>The answer is <strong>NO</strong>. Any four times punctured sphere admits two involutions, the quotient by which is an orbifold of the signature you describe. Similarly, you can take a punctured torus, and the quotient by the elliptic involution is one of your surfaces. Conversely, you can cover one of them by a torus or a four-times-punctured sphere. Since very few tori are arithmetic, same is true of your class of groups.</p>