For small , the known compact hyperbolic -orbifolds of minimal volume are intimately related to Coxeter groups of smallest rank. For and , these Coxeter groups are given by the triangle group and the tetrahedral group , and they are also distinguished by the fact that they have minimal growth rate among all cocompact hyperbolic Coxeter groups in , respectively. In this work, we consider the cocompact Coxeter simplex group with Coxeter symbol in and the cocompact Coxeter prism group based on in . Both groups are arithmetic and related to the fundamental group of the minimal volume arithmetic compact hyperbolic -orbifold for and , respectively. Here, we prove that the group is distinguished by having smallest growth rate among all Coxeter groups acting cocompactly on for and , 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. 725–741DOI 10.4171/GGD/663