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 H n , n ≥ 2, we show that (a) one can produce infinitely many maximal quasi-arithmetic reflection groups acting on H 2 ; (b) they admit infinitely many different fields of definition; (c) the degrees of their fields of definition are unbounded. However, for n ≥ 14 the technique initially developed by Vinberg shows that there are still finitely many fields of definitions in the quasi-arithmetic case.