Fields of definition for triangle groups as Fuchsian groups
The compact hyperbolic triangle group $Δ(p,q,r)$ 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.
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