Bartholdi, Laurent
Neuhauser, Markus
Woess, Wolfgang
Horocyclic products of trees
Journal Of The European Mathematical Society
Journal Of The European Mathematical Society
Journal Of The European Mathematical Society
Journal Of The European Mathematical Society
10
restricted wreath product
trees
horocycles
Diestel-Leader graph
growth function
normal form
Markov operator
spectrum
Diestel-Leader Graphs
Finitely Presented Group
Arc-Transitive Digraphs
Random-Walks
Equilateral Triangle
Lamplighter Groups
Infinite-Graphs
Affine Group
Boundary
2008
2008
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.
Journal Of The European Mathematical Society
Journal Articles
10.4171/JEMS/130