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On growth types of quotients of Coxeter groups by parabolic subgroups

Sankaran Viswanath
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Abstract

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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Sankaran Viswanath. 2006-03-01. On growth types of quotients of Coxeter groups by parabolic subgroups. https://arxiv.org/abs/math/0601482

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