Horocyclic products of trees

Laurent Bartholdi ; Markus Neuhauser ; Wolfgang Woess

Journal of the European Mathematical Society, Tome 10 (2008) no. 3, pp. 771-816

Résumé

Let T1​,...,Td​ be homogeneous trees with degrees q1​+1,...,qd​+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 T1​,...,Td​ is the graph DL(q1​,...,qd​) consisting of all d-tuples x1​⋯xd​∈T1​×⋯×Td​ with h(x1​)+⋯+h(xd​)=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 q1​=q2​=q then we obtain a Cayley graph of the lamplighter group (wreath product) Zq​≀Z. If d=3 and q1​=q2​=q3​=q then DL is the Cayley graph of a finitely presented group into which the lamplighter group embeds naturally. Also when d≥4 and q1​=⋯=qd​=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 qj​ do not all coincide, DL(q1​,...,qd​) 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 l2-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.

Détail

DOI : 10.4171/jems/130

Classification : 05-XX, 20-XX, 00-XX
Keywords: Restricted wreath product, trees, horocycles, Diestel–Leader graph, growth function, normal form, Markov operator, spectrum