Diestel-Leader群的度量性质
Metric Properties of Diestel-Leader Groups
摘要：
本文研究了Cayley图是Diestel-Leader图$DL_d(q)$的群$Γ_d(q)$关于给定生成集$S_{d,q}$的度量性质.这些群体提供了一个几何推广的家庭的点灯人群体，其凯莱图相对于一定的生成集是Diestel-Leader图$DL_2(q)$。Bartholdi、Neuhauser和Woess在\cite{BNW}中指出，对于$d \geq 3$，$Γ_d(q)$属于$F_{d-1}$型，而不是$F_d$型。我们在下面表明，这些群有关于生成集$S_{d,q}$的任意深度的死端元素，以及无穷多个锥类型，因此没有测地线的正则语言。这些结果证明了使用组合公式来计算相对于$S_{d,q}$的群元素的字长，这也证明了在该文件中，依赖于Diestel-Leader图的几何形状。
英文摘要：
In this paper we investigate metric properties of the groups $Γ_d(q)$ 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 $d \geq 3$, $Γ_d(q)$ 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.
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