Hyperbolic Coxeter groups and minimal growth rates in dimensions four and five
Abstract
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 $\hbox{Isom}\mathbb H^n$, respectively. In this work, we consider the cocompact Coxeter simplex group $G_4$ with Coxeter symbol $[5,3,3,3]$ in $\hbox{Isom}\mathbb H^4$ and the cocompact Coxeter prism group $G_5$ based on $[5,3,3,3,3]$ in $\hbox{Isom}\mathbb H^5$. 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 $G_n$ is distinguished by having smallest growth rate among all Coxeter groups acting cocompactly on $\mathbb H^n$ 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.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2020
- DOI:
- 10.48550/arXiv.2008.10961
- arXiv:
- arXiv:2008.10961
- Bibcode:
- 2020arXiv200810961B
- Keywords:
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- Mathematics - Geometric Topology;
- Mathematics - Combinatorics;
- 20F55;
- 26A12;
- 22E40;
- 11R06
- E-Print:
- Version 3 is the final version accepted for publication in Groups, Geometry and Dynamics