Fields of definition for triangle groups as Fuchsian groups

TL;DR

This paper classifies compact hyperbolic triangle groups with specific algebraic properties, identifying exactly eleven such groups and confirming no others exist over totally real fields, thus resolving several longstanding conjectures.

Contribution

It proves the finiteness of compact hyperbolic triangle groups conjugate to subgroups of PSL_2 over their trace fields and confirms no others exist over totally real fields.

Findings

Exactly eleven such groups are conjugate to subgroups of PSL_2(K).

No additional groups are conjugate to subgroups of PSL_2(L) for any totally real field L.

Resolved five recent conjectures of McMullen.

Abstract

The compact hyperbolic triangle group $\Delta(p,q,r)$ 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 \[K = \mathbf{Q}(\cos(\pi/p), \cos(\pi/q), \cos(\pi/r)).\] 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.