Horocyclic products of trees

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 $x_1\cdots x_d\in T_1\times \cdots \times T_d$ 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 $q_1=\cdots =q_d=q$ 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 \( \DL \).