I study geometric actions of groups on Riemannian manifolds and orbifolds. More precisely, let G be a discrete group of isometries of the hyperbolic Lobachevsky space H^n, then G is a hyperbolic lattice if the quotient hyperbolic orbifold H^n/G has finite volume. If G acts freely (ie is torsion free) then H^n/G is a hyperbolic manifold. If G is an arithmetic or quasi-arithmetic lattice then H^n/G is called so. Various examples of hyperbolic lattices are provided by the theory of hyperbolic reflection groups developed by Vinberg in 1967. A natural fundamental domain of a hyperbolic lattice generated by reflections in hyperplanes is a Coxeter polytope. In his fundamental paper, Vinberg described hyperbolic Coxeter polytopes in terms of its Coxeter diagrams (graphs) and proved a (quasi-)arithmeticity criterion for a hyperbolic reflection group.
The modern study of discrete groups combines:
There are a lot of natural open questions in this theory.