For small n, the known compact hyperbolic n-orbifolds of minimal volume are intimately related to Coxeter groups of smallest rank. For n=2 and 3, these Coxeter groups are given by the triangle group [7,3] and the tetrahedral group [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, respectively. In this work, we consider the cocompact Coxeter simplex group G4 with Coxeter symbol [5,3,3,3] in IsomH4 and the cocompact Coxeter prism group G5 based on [5,3,3,3,3] in IsomH5. Both groups are arithmetic and related to the fundamental group of the minimal volume arithmetic compact hyperbolic n-orbifold for n=4 and 5, respectively. Here, we prove that the group Gn is distinguished by having smallest growth rate among all Coxeter groups acting cocompactly on Hn for n=4 and 5, 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.
Naomi Bredon; Ruth Kellerhals. Hyperbolic Coxeter groups and minimal growth rates in dimensions four and five. Groups, geometry, and dynamics, Tome 16 (2022) no. 2, pp. 725-741. doi: 10.4171/ggd/663
@article{10_4171_ggd_663,
author = {Naomi Bredon and Ruth Kellerhals},
title = {Hyperbolic {Coxeter} groups and minimal growth rates in dimensions four and five},
journal = {Groups, geometry, and dynamics},
pages = {725--741},
year = {2022},
volume = {16},
number = {2},
doi = {10.4171/ggd/663},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/663/}
}
TY - JOUR
AU - Naomi Bredon
AU - Ruth Kellerhals
TI - Hyperbolic Coxeter groups and minimal growth rates in dimensions four and five
JO - Groups, geometry, and dynamics
PY - 2022
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EP - 741
VL - 16
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.4171/ggd/663/
DO - 10.4171/ggd/663
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%0 Journal Article
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%J Groups, geometry, and dynamics
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