Geodesic
Reflection Subgroups of Reflection Groups
Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 4, pp. 90-92

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Let $G$ be a discrete group generated by reflections in hyperbolic or Euclidean space, and let $H\subset G$ be a finite index reflection subgroup. Suppose that the fundamental chamber of $G$ is a finite volume polytope with $k$ facets. We prove that the fundamental chamber of $H$ has at least $k$ facets.
Keywords: reflection group, Coxeter polytope. 
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     author = {P. V. Tumarkin and A. A. Felikson},
     title = {Reflection {Subgroups} of {Reflection} {Groups}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {90--92},
     publisher = {mathdoc},
     volume = {38},
     number = {4},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2004_38_4_a10/}
}
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P. V. Tumarkin; A. A. Felikson. Reflection Subgroups of Reflection Groups. Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 4, pp. 90-92. http://geodesic.mathdoc.fr/item/FAA_2004_38_4_a10/