# Horocyclic products of trees

### Wolfgang Woess

Technische Universität Graz, Austria### Markus Neuhauser

Universität Wien, Austria### Laurent Bartholdi

Georg-August-Universität Göttingen, Germany

## Abstract

Let $T_{1},…,T_{d}$ be homogeneous trees with degrees $q_{1}+1,…,q_{d}+1≥3,$ respectively. For each tree, let $h:T_{j}→Z$ be the Busemann function with respect to a fixed boundary point (end). Its level sets are the horocycles. The horocyclic product of $T_{1},…,T_{d}$ is the graph $DL(q_{1},…,q_{d})$ consisting of all $d$-tuples $x_{1}⋯x_{d}∈T_{1}×⋯×T_{d}$ with $h(x_{1})+⋯+h(x_{d})=0$, 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) $Z_{q}≀Z$. 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 $d≥4$ and $q_{1}=⋯=q_{d}=q$ 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, $DL(q_{1},…,q_{d})$ 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 $ℓ_{2}$-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 $DL$.

## Cite this article

Wolfgang Woess, Markus Neuhauser, Laurent Bartholdi, Horocyclic products of trees. J. Eur. Math. Soc. 10 (2008), no. 3, pp. 771–816

DOI 10.4171/JEMS/130