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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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/

[1] Andreev E. M., Matem. zametki, 8 (1970), 521–527 | MR | Zbl

[2] Coxeter H. S. M., “Finite groups generated by reflections, and their subgroups generated by reflections”, Proc. Cambridge Philos. Soc., 30 (1934), 466–482 | DOI | MR | Zbl