Triangle groups - uniqueness and trace field

2010-10-19T14:29:42Z

Dear all,

again I need your help for the following two questions: Suppose we have a triangle group of signature (p,q,\infty).

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?

Thank you very much for your answer Ali

Answer by HW for Triangle groups - uniqueness and trace field

2010-10-22T21:03:22Z

One page 159 of The Arithmetic of Hyperbolic Manifolds by Maclachlan and Reid:-

In more detail, suppose that $\Gamma$ is a $(l,m,n)$-triangle group where $1/l+1/m+1/n<1$ so that $\Gamma$ has the presentation

$\langle x,y\mid x^l=y^m=(xy)^n=1 \rangle$.

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).

More generally, Maclachlan and Reid should have the answers to all your questions about trace fields.

Triangle groups - uniqueness and trace field - MathOverflow

Triangle groups - uniqueness and trace field

Answer by HW for Triangle groups - uniqueness and trace field