Vinberg’s theorem on hyperbolic reflection groups

In this talk we will expalin the main ideas of the proof of the following theorem of Vinberg: Let f be an integral quadratic form of signature (n, 1). If n ≥ 30 then the subgroup of SO(n, 1)(Z) which is generated by all hyperbolic reflections has infinite index. As a consequence of this theorem we will show that certain hypergeometric monodromy groups are thin.