On the non-amenability of the reflective quotient \Rmnum{1}: The rational case

TL;DR

This paper generalizes Vinberg's theorem, showing that for quadratic forms of signature (n,1) with n > 91, the reflective quotient contains a non-abelian free group, indicating non-amenability.

Contribution

It extends Vinberg's result by proving non-amenability of the reflective quotient for higher dimensions (n > 91) in the rational case.

Findings

For n > 91, the reflective quotient contains a non-abelian free group.

The non-amenability of the quotient is established in the rational case.

Generalization of Vinberg's theorem to higher dimensions.

Abstract

Let O(f,Z) be the integral orthogonal group of an integral quadratic form f of signature (n,1). Let R(f,Z) be the subgroup of O(f,Z) generated by all hyperbolic reflections. Vinberg proved that if n > 29 then the reflective quotient O(f,Z)/R(f,Z) is infinite. In this note we generalize Vinberg's theorem and prove that if n > 91 then O(f,Z)/R(f,Z) contains a non-abelian free group (and thus it is not amenable).