Bounds for discrete hyperbolic arithmetic reflection groups in dimension 2

Abstract

Here we show that if Γ is an arithmetic Fuchsian group of genus 0, then the totally real defining field k of Γ must be such that [k : ℚ] ⩽ 11. The same inequality holds for discrete arithmetic hyperbolic reflection groups acting on a two-dimensional hyperbolic space ℍ2. In addition, we show that there exists an arithmetic Fuchsian group of genus 0 containing an element of order N if and only if N ∈ {2, 3, …, 16, 18, 20, 22, 24, 26, 28, 30, 36}. A slightly less precise statement holds for discrete arithmetic hyperbolic reflection groups acting on ℍ2.