论文标题
双曲线coxeter组和尺寸最小的增长率四和五
Hyperbolic Coxeter groups and minimal growth rates in dimensions four and five
论文作者
论文摘要
对于小$ n $，已知的紧凑型双曲线$ n $ -Orbifolds的最小量与最小排名的Coxeter组密切相关。对于$ n = 2 $和$ 3 $，这些Coxeter组由三角形组$ [7,3] $和四面体组$ [3,5,3] $给出，并且它们也以所有共同型双胞胎coxeter组在$ \ hbox {Isom} {isom} {isom} {isom} \ nath $ nytical crompact双bolic coxeter组中的增长率最低而得到区别。在这项工作中，我们考虑了CoCompact Coxeter Simplex $ G_4 $与Coxeter符号$ [5,3,3,3] $ [5,3,3,3] $中的$ \ hbox {isom} \ Mathbb H^4 $和CooCompact Coxeter Prism $ g_5 $ g_5 $ g_5 $ g_5 $ g_5 $基于$ [5,3,3,3,3,3,3] $ in $ \ hbox $ \ hbox} $ \ hbox}这两个组都是算术的，并且与最小量的算术算术紧凑型双曲线$ n $ -Orbifold分别分别为$ n = 4 $和$ 5 $有关。在这里，我们证明了组$ g_n $的区别是，在$ n = 4 $和$ 5 $上分别在$ \ mathbb h^n $上表现出的所有Coxeter组的增长率最小。该证明是基于紧凑型双曲线甲壳虫多面体的组合特性，一些部分分类结果以及相关coxeter基团生长速率的某些单调性能。
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.
