Horocyclic products of trees

Bartholdi, Laurent, Neuhauser, Markus, and Woess, Wolfgang. "Horocyclic products of trees." Journal of the European Mathematical Society 010.3 (2008): 771-816. <http://eudml.org/doc/277629>.

@article{Bartholdi2008,
abstract = {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 $\mathfrak \{h\}:T_j\rightarrow \mathbb \{Z\}$ 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 $\mathsf \{DL\}(q_1,\dots ,q_d)$ consisting of all $d$-tuples $x_1\cdots x_d \in T_1\times \dots \times T_d$ with $\mathfrak \{h\}(x_1)+\dots +\mathfrak \{h\}(x_d)=0$, 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) $\mathfrak \{Z\}_q\wr \mathfrak \{Z\}$. If $d=3$ and $q_1=q_2=q_3=q$ then $\mathsf \{DL\}$ is the Cayley graph of a finitely presented group into which the lamplighter group embeds naturally. Also when $d\ge 4$ 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 $\mathsf \{DL\}$ 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, $\mathsf \{DL\}(q_1,\dots ,q_d)$ 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 $\mathsf \{DL\}$ 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 $\mathsf \{DL\}$. It coincides with a part of the geometric boundary of $\mathsf \{DL\}$.},
author = {Bartholdi, Laurent, Neuhauser, Markus, Woess, Wolfgang},
journal = {Journal of the European Mathematical Society},
keywords = {restricted wreath product; trees; horocycles; Diestel–Leader graph; growth function; normal form; Markov operator; spectrum; restricted wreath product; trees; horocycles; Diestel-Leader graph; growth function; normal form; Markov operator; spectrum},
language = {eng},
number = {3},
pages = {771-816},
publisher = {European Mathematical Society Publishing House},
title = {Horocyclic products of trees},
url = {http://eudml.org/doc/277629},
volume = {010},
year = {2008},
}

TY - JOUR
AU - Bartholdi, Laurent
AU - Neuhauser, Markus
AU - Woess, Wolfgang
TI - Horocyclic products of trees
JO - Journal of the European Mathematical Society
PY - 2008
PB - European Mathematical Society Publishing House
VL - 010
IS - 3
SP - 771
EP - 816
AB - 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 $\mathfrak {h}:T_j\rightarrow \mathbb {Z}$ 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 $\mathsf {DL}(q_1,\dots ,q_d)$ consisting of all $d$-tuples $x_1\cdots x_d \in T_1\times \dots \times T_d$ with $\mathfrak {h}(x_1)+\dots +\mathfrak {h}(x_d)=0$, 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) $\mathfrak {Z}_q\wr \mathfrak {Z}$. If $d=3$ and $q_1=q_2=q_3=q$ then $\mathsf {DL}$ is the Cayley graph of a finitely presented group into which the lamplighter group embeds naturally. Also when $d\ge 4$ 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 $\mathsf {DL}$ 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, $\mathsf {DL}(q_1,\dots ,q_d)$ 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 $\mathsf {DL}$ 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 $\mathsf {DL}$. It coincides with a part of the geometric boundary of $\mathsf {DL}$.
LA - eng
KW - restricted wreath product; trees; horocycles; Diestel–Leader graph; growth function; normal form; Markov operator; spectrum; restricted wreath product; trees; horocycles; Diestel-Leader graph; growth function; normal form; Markov operator; spectrum
UR - http://eudml.org/doc/277629
ER -