Coxeter decompositions of hyperbolic simplexes

A Coxeter decomposition of a polyhedron in a hyperbolic space $\mathbb H^n$ is a decomposition of it into finitely many Coxeter polyhedra such that any two tiles having a common facet are symmetric with respect to it. The classification of Coxeter decompositions is closely related to the problem of the classification of finite-index subgroups generated by reflections in discrete hyperbolic groups generated by reflections. All Coxeter decompositions of simplexes in the hyperbolic spaces $\mathbb H^n$ with $n>3$ are described in this paper.