Infinitely many quasi-arithmetic maximal reflection groups

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.