Horocyclic products of trees

Let T-1, ... , T-d be homogeneous trees with degrees q(1) + 1, ... , q(d) + 1 >= 3; respectively. For each tree, let h : Tj -> 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(q1, ... , q(d)) consisting of all d-tuples x(1) ... x(d) is an element of T-1 x ... x 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) 3q (sic) Z. If d = 3 and q(1) = q(2) = q(3) = q then DL is a Cayley graph of a finitely presented group into which the lamplighter group embeds naturally. In general, 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 Fd - 1, but not Fd. 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 a 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.