BOUNDS FOR ARITHMETIC HYPERBOLIC REFLECTION GROUPS IN DIMENSION 2

Abstract

The work of Nikulin and Agol, Belolipetsky, Storm, and Whyte shows that only finitely many number fields may serve as the field of definition of an arithmetic hyperbolic reflection group. An important problem posed by Nikulin is to enumerate these fields and their degrees. In this paper we prove that in dimension 2 the degree of these fields is at most 7. More generally we prove that the degree of the field of definition of the quaternion algebra associated to an arithmetic Fuchsian group of genus 0 is at most 7, confirming a conjecture of Long, Maclachlan and Reid. We also obtain upper bounds for the discriminants of these fields of definition, allowing for an enumeration which should be useful for the classification of arithmetic hyperbolic reflection groups.