Galois orbits in the moduli space of all triangles

Abstract

Every $$ in the torus $$ determines a unique spherical, Euclidean or hyperbolic triangle $$ with angles $$. In this paper we study the Galois orbits $$ of torsion points $$, focusing on the ramification density $$ We show that the closure $$ of the set of values of $$ is a countable subset of $$, with 0 and 1 as isolated points. The spectral gaps at 0 and 1 lead to general finiteness statements for the classical triangle groups $$. For example, we obtain a conceptual proof, based on equidistribution, that the set of arithmetic triangle groups is finite.