Metric Properties of Diestel-Leader Groups
Metric Properties of Diestel-Leader Groups
复制标题
Diestel-Leader 组的公制特性
DOI：
10.1307/mmj/1370870377
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
J. Taback
中科院分区：
文献类型：
--
作者：
M. Stein;J. Taback
本文研究了Cayley图为Diestel-Leader图的群$\Gamma_d(Q)$关于给定生成集$S_(Q)$的度量性质。这些群是照明族的几何推广，其关于某一生成集的Cayley图是Diestel-Leader图$DL_2(Q)$。Bartholdi，Neuhauser和Woess在Cite{BNW}中证明了对于$dgeq3$，$\Gamma_d(Q)$是$F_(d-1)$类型，而不是$F_d$。我们证明了这些群具有关于生成集$S_{d，q}$的任意深度的死端元素，以及无限多个锥形类型，因此没有正则测地线语言。利用关于$S的群元字长的一个组合公式证明了这些结果，该公式依赖于Diestel-Leader图的几何.
In this paper we investigate metric properties of the groups $\Gamma_d(q)$ whose Cayley graphs are the Diestel-Leader graphs $DL_d(q)$ with respect to a given generating set $S_{d,q}$. These groups provide a geometric generalization of the family of lamplighter groups, whose Cayley graphs with respect to a certain generating set are the Diestel-Leader graphs $DL_2(q)$. Bartholdi, Neuhauser and Woess in \cite{BNW} show that for $d \geq 3$, $\Gamma_d(q)$ is of type $F_{d-1}$ but not $F_d$. We show below that these groups have dead end elements of arbitrary depth with respect to the generating set $S_{d,q}$, as well as infinitely many cone types and hence no regular language of geodesics. These results are proven using a combinatorial formula to compute the word length of group elements with respect to $S_{d,q}$ which is also proven in the paper and relies on the geometry of the Diestel-Leader graphs.