# 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 \( \hor:T_j \to \Z \) 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 \( \DL(q_1,\dots,q_d) \) consisting of all $d$-tuples $x_{1}⋯x_{d}∈T_{1}×⋯×T_{d}$ 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) \( \Zq \wr \Z \). 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, \( \DL(q_1,\dots,q_d) \) 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 $ℓ_{2}$-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