91É«ÇéƬ

Abstract

The study of triangle groups \(\Delta(p,q,r) = \langle x,y,z | x^p, y^q, z^r, xyz \rangle\) dates back to the 19th century and work of Schwarz, Poincar\'e, and Fricke. If

\[\frac{1}{p} + \frac{1}{q} + \frac{1}{r}Ìý < 1,\]

then \(\Delta\) can be realized as the symmetry group of a tiling of the hyperbolic plane by hyperbolic triangles, which allows one to construct a faithful discrete representation

\[\Delta \rightarrow \mathrm{PSL}_2(\mathbf{R}).\]

The rigidity of the situation implies that the image actually lands inside \(\mathrm{PSL}_2(K)\) for some number field \(K \hookrightarrow \mathbf{R}\). The question we address in this talk is: when can we take \(K\) to be a \emph{totally} real field? This question was first addressed directly by Maclachlan and Waterman in 1985, and more recently by Curt McMullen. In this talk, we completely resolve the question, resolving a series of conjectures by McMullen. This is joint work with Qiankang Chen.