Horocyclic products of trees
Horocyclic products of trees
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树木的环环产物
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发表时间：
2006
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通讯作者：
W. Woess
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作者：
L. Bartholdi;M. Neuhauser;W. Woess
设$T_1，dots，Td $分别是度为$q_1+1，dots，qd +1ge 3，$的齐次树.对于每棵树，设$hor：T_j O $是相对于固定边界点（端点）的Busemann函数。它的水平集是horocycles。图$DL（q_1，dots，q_d）$由T_1中的所有$d$-元组$x_1cdotsx_d组成 艾姆斯圆点 当h（x_1）+dots+hor（x_d）=0时，T_d$具有自然的邻域关系。在本文中，我们探讨了这些图及其等距群的几何，代数，分析和概率性质。若d=2且q_1 =q_2=q，则得到点灯群（圈积）的Cayley图q wr $.若d = 3且q_1 = q_2 = q_3 = q$，则DL$是点灯人群自然嵌入其中的一个二次表示群的Cayley图.并且当$dge 4$和$q_1 = dots = q_d = q$使得$q$的分解中的每一个素幂都大于$d-1$时，我们证明了$DL$是一个n-表示群的Cayley图。此组的类型为$F_{d-1}$，但不是$F_d$。它不是自动的，但在大多数情况下它是一个自动机组。另一方面，当$q_j$不完全重合时，$DL（q_1，dots，q_d）$是一个点传递图，但不是一个点生成群的Cayley图.事实上，它甚至不承认具有多个轨道和有限点稳定器的群作用。$DL$上的"简单随机游走“算子的$ell^2$-谱总是纯点。当$d=2$时，它从以前的工作中显式地知道，而对于$d=3$，我们显式地计算它。最后，我们确定了$DL$上的一大类群不变随机游动的Poisson边界。它与$DL$的部分几何边界重合。
Let $T_1,dots, T_d$ be homogeneous trees with degrees $q_1+1, dots, q_d+1 ge 3,$ respectively. For each tree, let $hor:T_j o $ be the Busemann function with respect to a fixed boundary point (end). Its level sets are the horocycles. The horocyclic product of $T_1,dots, T_d$ is the graph $DL(q_1,dots,q_d)$ consisting of all $d$-tuples $x_1 cdots x_d in T_1 imes dots imes T_d$ with $hor(x_1)+dots+hor(x_d)=0$, equipped with a natural neighbourhood relation. In the present paper, we explore the geometric, algebraic, analytic and probabilistic properties of these graphs and their isometry groups. If $d=2$ and $q_1=q_2=q$ then we obtain a Cayley graph of the lamplighter group (wreath product) $q wr $. If $d = 3$ and $q_1 = q_2 = q_3 = q$ then $DL$ is the Cayley graph of a finitely presented group into which the lamplighter group embeds naturally. Also when $dge 4$ and $q_1 = dots = q_d = q$ is such that each prime power in the decomposition of $q$ is larger than $d-1$, we show that $DL$ is a Cayley graph of a finitely presented group. This group is of type $F_{d-1}$, but not $F_d$. It is not automatic, but it is an automata group in most cases. On the other hand, when the $q_j$ do not all coincide, $DL(q_1,dots,q_d)$ is a vertex-transitive graph, but is not the Cayley graph of a finitely generated group. Indeed, it does not even admit a group action with finitely many orbits and finite point stabilizers. The $ell^2$-spectrum of the ``simple random walk' operator on $DL$ is always pure point. When $d=2$, it is known explicitly from previous work, while for $d=3$ we compute it explicitly. Finally, we determine the Poisson boundary of a large class of group-invariant random walks on $DL$. It coincides with a part of the geometric boundary of $DL$.