Triangle groups - uniqueness and trace field - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T09:04:28Z http://mathoverflow.net/feeds/question/42794 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/42794/triangle-groups-uniqueness-and-trace-field Triangle groups - uniqueness and trace field Ali K 2010-10-19T14:29:42Z 2010-10-22T21:03:22Z <p>Dear all,</p> <p>again I need your help for the following two questions: Suppose we have a triangle group of signature (p,q,\infty). </p> <p>1) When is such a group unique (up to isomorphism)? 2) Do you have a method how to calculate the trace field of such a group? Is the trace field unique?</p> <p>Thank you very much for your answer Ali</p> http://mathoverflow.net/questions/42794/triangle-groups-uniqueness-and-trace-field/43233#43233 Answer by HW for Triangle groups - uniqueness and trace field HW 2010-10-22T21:03:22Z 2010-10-22T21:03:22Z <p>One page 159 of <em>The Arithmetic of Hyperbolic Manifolds</em> by Maclachlan and Reid:-</p> <blockquote> <p>In more detail, suppose that $\Gamma$ is a $(l,m,n)$-triangle group where $1/l+1/m+1/n&lt;1$ so that $\Gamma$ has the presentation</p> <p>$\langle x,y\mid x^l=y^m=(xy)^n=1 \rangle$.</p> <p>Then $\mathbb{Q}(\mathrm{tr}\Gamma)=\mathbb{Q}(\cos\pi/l,\cos\pi/m,\cos\pi/n)$ (see (3.25)), and the invariant trace field is a subfield of this totally real number field (see Exercise 4.9, No. 1).</p> </blockquote> <p>More generally, Maclachlan and Reid should have the answers to all your questions about trace fields.</p>