On growth types of quotients of Coxeter groups by parabolic subgroups

Details On growth types of quotients of Coxeter groups by parabolic subgroups On growth types of quotients of Coxeter groups by parabolic subgroups

2006-01-19

arxiv.org/abs/math/0601482v2

Mathematics

Group Theory

10 pages; The exposition has been made more concise and an additional proposition is proved in the final section

Scientific paper

The principal objects studied in this note are Coxeter groups $W$ that are neither finite nor affine. A well known result of de la Harpe asserts that such groups have exponential growth. We consider quotients of $W$ by its parabolic subgroups and by a certain class of reflection subgroups. We show that these quotients have exponential growth as well. To achieve this, we use a theorem of Dyer to construct a reflection subgroup of $W$ that is isomorphic to the universal Coxeter group on three generators. The results are all proved under the restriction that the Coxeter diagram of $W$ is simply laced, and some remarks made on how this restriction may be relaxed.

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