Horocyclic products of trees (Q936156)

Summary: Let \(T_{1},\ldots ,T_{d}\) be homogeneous trees with degrees \(q_{1}+1,\ldots ,q_{d}+1 \geq 3,\) 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 \(T_{1},\ldots ,T_{d}\) is the graph \(\mathsf{DL}(q_{1},\ldots ,q_{d})\) consisting of all \(d\)-tuples \(x_{1} \cdots x_{d} \in T_{1} \times \cdots \times T_{d}\) with \({\mathfrak h}(x_{1})+\cdots+{\mathfrak h}(x_{d}) = 0\), 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) \(\mathcal {3}_{q} \wr \mathbb Z\). If \(d = 3\) and \(q_{1} = q_{2} = q_{3} = q\) then \textsf{DL} is the Cayley graph of a finitely presented group into which the lamplighter group embeds naturally. Also when \(d \geq 4\) 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, \(\mathsf{DL}(q_{1},\ldots,q_{d})\) 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 \textsf{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 \textsf{DL}. It coincides with a part of the geometric boundary of \textsf{DL}.