English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

For small $n$, the known compact hyperbolic $n$-orbifolds of minimal volumeare 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 thetetrahedral group $[3,5,3]$, and they are also distinguished by the fact thatthey have minimal growth rate among all cocompact hyperbolic Coxeter groups in$\hbox{Isom}\mathbb H^n$, respectively. In this work, we consider the cocompactCoxeter 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 andrelated to the fundamental group of the minimal volume arithmetic compacthyperbolic $n$-orbifold for $n=4$ and $5$, respectively. Here, we prove thatthe group $G_n$ is distinguished by having smallest growth rate among allCoxeter groups acting cocompactly on $\mathbb H^n$ for $n=4$ and $5$,respectively. The proof is based on combinatorial properties of compacthyperbolic Coxeter polyhedra, some partial classification results and certainmonotonicity properties of growth rates of the associated Coxeter groups.

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