Fields of definition for triangle groups as Fuchsian groups

Abstract

The compact hyperbolic triangle group $\hat{I}”(p,q,r)$ admits a canonical representation to ${\mathrm{PSL}}_2(\mathbf{R})$ with discrete image which is unique up to conjugation. The trace field of this representation is \[K = \mathbf{Q}(\cos(π/p), \cos(π/q), \cos(π/r)).\] We prove that there are exactly eleven such groups which are conjugate to subgroups of ${\mathrm{PSL}}_2(K)$. Moreover, we prove that there are no additional compact hyperbolic triangle groups which are conjugate to subgroups of ${\mathrm{PSL}}_2(L)$ for any totally real field $L$. This answers a question first raised by Waterman and Machlachlan, and also resolves (in the positive) five (interrelated) recent conjectures of McMullen.

40 pages, comments welcome