Horocyclic products of trees

Abstract

Abstract. Let T 1 , . . . , T d be homogeneous trees with degrees q 1 +1, . . . , q d +1 ≥ 3, respectively. For each tree, let h : T j → Z be the Busemann function with respect to a fixed boundary point (end). Its level sets are the horocycles. The horocyclic product ofequipped 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) Z q 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 F d−1 , but not F d . 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 vertextransitive 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 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.

On the spectrum of lamplighter groups and percolation clusters

“…One reason can be found in the fact that H Z has an inherent structure of self-similarity, compare with Bartholdi et al [2]. Later, further classes of groups and convolution operators on them with pure point spectrum were found by Bartholdi et al [3]. The origin of this paper was the question at which level of generality a pure point spectrum occurs for typical lamplighter random walks on general wreath products H G.…”

Sharply k-arc-transitive-digraphs: Finite and infinite examples

“…The Diestel-Leader graphs DL(p, q), for p, q ≥ 2 were originally defined in [5]. The Diestel-Leader graphs can be described in various ways; one description is in [11,Example 1] and an another one in [1]. In [1] they are described in terms of the horocyclic product of trees.…”

“…The Diestel-Leader graphs can be described in various ways; one description is in [11,Example 1] and an another one in [1]. In [1] they are described in terms of the horocyclic product of trees. The direct fibre product in [13] is an analogue of the horocyclic product and one can describe the Diestel-Leader graphs in terms of the direct fibre product, as explained in [13,Section 4.5].…”

The Dehn function of Baumslag’s metabelian group

“…The description of a Cayley graph of m in [3] as a horocyclic product is essentially the same as a horosphere corresponding to a barycentric ray in the product of the three trees. (It is contained in the horosphere and it is at finite Hausdorff distance from it.)…”

“…These groups appear in other guises: both and m are groups of affine matrices and have Cayley graphs that are horocyclic products of trees [3]; for p prime, p is a cocompact lattice in Sol 5 (F p ((t))) [12].…”