On the non-amenability of the reflective quotient

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 [Vi3] proved that if n ≥ 30 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 ≥ 92 then O(f, ℤ)/R(f, ℤ) contains a non-abelian free group (and thus it is not amenable).