Hyperbolic Coxeter groups and minimal growth rates in dimensions four and five

  • Naomi Bredon

    University of Fribourg, Switzerland
  • Ruth Kellerhals

    University of Fribourg, Switzerland
Hyperbolic Coxeter groups and minimal growth rates in dimensions four and five cover
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Abstract

For small nn, the known compact hyperbolic nn-orbifolds of minimal volume are intimately related to Coxeter groups of smallest rank. For n=2n=2 and 33, these Coxeter groups are given by the triangle group [7,3][7,3] and the tetrahedral group [3,5,3][3,5,3], and they are also distinguished by the fact that they have minimal growth rate among all cocompact hyperbolic Coxeter groups in IsomHn\operatorname{Isom}\mathbb H^n, respectively. In this work, we consider the cocompact Coxeter simplex group G4G_4 with Coxeter symbol [5,3,3,3][5,3,3,3] in IsomH4\operatorname{Isom}\mathbb H^4 and the cocompact Coxeter prism group G5G_5 based on [5,3,3,3,3][5,3,3,3,3] in IsomH5\operatorname{Isom}\mathbb H^5. Both groups are arithmetic and related to the fundamental group of the minimal volume arithmetic compact hyperbolic nn-orbifold for n=4n=4 and 55, respectively. Here, we prove that the group GnG_n is distinguished by having smallest growth rate among all Coxeter groups acting cocompactly on Hn\mathbb H^n for n=4n=4 and 55, respectively. The proof is based on combinatorial properties of compact hyperbolic Coxeter polyhedra, some partial classification results and certain monotonicity properties of growth rates of the associated Coxeter groups.

Cite this article

Naomi Bredon, Ruth Kellerhals, Hyperbolic Coxeter groups and minimal growth rates in dimensions four and five. Groups Geom. Dyn. 16 (2022), no. 2, pp. 725–741

DOI 10.4171/GGD/663