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 ${\mathbb{H}}^n$, $n\ge 2$, 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 $n\ge 14$ an approach initially developed by Vinberg shows that there are still finitely many fields of definitions in the quasi-arithmetic case.

8 pages, 2 figures; to appear in Proc. Amer. Math. Soc. (2022)