Galois orbits in the moduli space of all triangles

Abstract

Every 𝑎 in the torus 𝐴 = ℝ³/2ℤ³ determines a unique spherical, Euclidean or hyperbolic triangle 𝑇(𝑎) with angles (𝜋 𝑎_𝑖). In this paper we study the Galois orbits Gal(𝑎) of torsion points 𝑎 ∈ 𝐴, focusing on the ramification density

𝜌(𝑎) = \frac{|{ 𝑏 ∈ Gal(𝑎) : 𝑇(𝑏) is spherical }|}{|Gal(𝑎)|}.

We show that the closure \overline{𝑅} of the set of values of 𝜌(𝑎) is a countable subset of [0, 1], with 0 and 1 as isolated points. The spectral gaps at 0 and 1 lead to general finiteness statements for the classical triangle groups Δ(𝑝, 𝑞, 𝑟) ⊂ SL₂(ℝ). For example, we obtain a conceptual proof, based on equidistribution, that the set of arithmetic triangle groups is finite.