Let be homogeneous trees with degrees respectively. For each tree, let be the Busemann function with respect to a fixed boundary point (end). Its level sets are the horocycles. The horocyclic product of is the graph consisting of all -tuples with , 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 and then we obtain a Cayley graph of the lamplighter group (wreath product) . If and then is the Cayley graph of a finitely presented group into which the lamplighter group embeds naturally. Also when and is such that each prime power in the decomposition of is larger than , we show that is a Cayley graph of a finitely presented group. This group is of type , but not . It is not automatic, but it is an automata group in most cases. On the other hand, when the do not all coincide, 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 -spectrum of the ``simple random walk'' operator on is always pure point. When , it is known explicitly from previous work, while for we compute it explicitly. Finally, we determine the Poisson boundary of a large class of group-invariant random walks on . It coincides with a part of the geometric boundary of .
Cite this article
Wolfgang Woess, Markus Neuhauser, Laurent Bartholdi, Horocyclic products of trees. J. Eur. Math. Soc. 10 (2008), no. 3, pp. 771–816DOI 10.4171/JEMS/130