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 \mathfrak h:T_j \to \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 \mathbb 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} .