Geodesic
Coxeter decompositions of hyperbolic simplexes
Sbornik. Mathematics, Tome 193 (2002) no. 12, pp. 1867-1888

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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. 
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     author = {A. A. Felikson},
     title = {Coxeter decompositions of hyperbolic simplexes},
     journal = {Sbornik. Mathematics},
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     number = {12},
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A. A. Felikson. Coxeter decompositions of hyperbolic simplexes. Sbornik. Mathematics, Tome 193 (2002) no. 12, pp. 1867-1888. http://geodesic.mathdoc.fr/item/SM_2002_193_12_a5/