Arithmeticity and commensurability classes of hyperbolic reflection groups

Hyperbolic reflection groups appear in various fields of mathematics such as algebraic geometry, discrete subgroups of Lie groups, geometric group theory, geometric topology, and number theory. Cofinite and cocompact hyperbolic reflection groups have the following feature: their fundamental domains are Coxeter polyhedra having simple geometric and combinatorial properties. Based on these properties, Vinberg initiated the theory of hyperbolic reflection groups and proved several fundamental and remarkable theorems related to their arithmeticity and commensurability classes. I will give a survey of this field, and will mention a few recent results in this direction. This talk is partially based on my work with S. Douba and J. Raimbault.