Horocyclic products of trees
نویسندگان
چکیده
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 graphDL(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 o 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 `-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.
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