# Horocyclic products of trees

### Wolfgang Woess

Technische Universität Graz, Austria### Markus Neuhauser

Universität Wien, Austria### Laurent Bartholdi

Georg-August-Universität Göttingen, Germany

## Abstract

Let $T_1,\dots, T_d$ be homogeneous trees with degrees $q_1+1, \dots, q_d+1 \ge 3,$ respectively. For each tree, let $\hor:T_j \to \Z$ be the Busemann function with respect to a fixed boundary point (end). Its level sets are the horocycles. The horocyclic product of $T_1,\dots, T_d$ is the graph $\DL(q_1,\dots,q_d)$ consisting of all $d$-tuples $x_1 \cdots x_d \in T_1 \times \dots \times T_d$ 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 $d\ge 4$ 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 $\DL$ 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 $\DL$.