Infinitely many quasi-arithmetic maximal reflection groups

TL;DR

This paper demonstrates the existence of infinitely many maximal quasi-arithmetic reflection groups in hyperbolic 2-space, contrasting with the finite nature of maximal arithmetic reflection groups, and explores their fields of definition.

Contribution

It establishes the existence of infinitely many such groups in $ ext{H}^2$ and analyzes their fields of definition, highlighting differences from the arithmetic case.

Findings

Infinitely many maximal quasi-arithmetic reflection groups in $ ext{H}^2$

These groups have infinitely many different fields of definition

Degrees of fields of definition are unbounded

Abstract

In contrast to the fact that there are only finitely many maximal arithmetic reflection groups acting on the hyperbolic space $\mathbb{H}^n$, $n\geq 2$, we show that: (a) one can produce infinitely many maximal quasi-arithmetic reflection groups acting on $\mathbb{H}^2$; (b) they admit infinitely many different fields of definition; (c) the degrees of their fields of definition are unbounded. However, for $n\geq 14$ an approach initially developed by Vinberg shows that there are still finitely many fields of definitions in the quasi-arithmetic case.