Proceedings of the American Mathematical Society

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:

1. one can produce infinitely many maximal quasi–arithmetic reflection groups acting on $\mathbb {H}^2$;
2. they admit infinitely many different fields of definition;
3. 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.