Let T_1, ..., T_d be homogeneous trees with branching numbers q_1, ..., q_d , respectively. For each tree, let h 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 consisting of all d-tuples x_1 ... x_d in the direct product of the trees with h(x_1)+...+h(x_d)=0, equipped with a natural neighbourhood relation. 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 of Z mod q with Z). If d = 3 and q_1 = q_2 = q_3 = q then we get the Cayley graph of a finitely presented group into which the lamplighter group embeds naturally. Also when d > 3 and q_1 ... = q_d = q is such that each prime power in the decomposition of q is larger than d-1, we get a Cayley graph of a finitely presented group. (Methods are inspired by "discrete subgroups of Lie groups.)
On the other hand, when the q_j do not all coincide, we get a vertex-transitive graph, but is not the Cayley graph of any 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 our graph is always pure point.
Finally, we determine the Poisson boundary of a large class of group-invariant random walks. It coincides with a part of the geometric boundary of the horocyclic product.
This is joint work with L. Bartholdi and M. Neuhauser.
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After the seminar we will take the speaker to lunch.
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James East james.east@sydney.edu.au