Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Horocyclic products of trees   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 of T_1,...,T_d is the graph DL(q_1,...,q_d) consisting of
all d-tuples x_1...x_d in T_1x...xT_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) (Z/qZ) wr Z. If d=3 and q_1=q_2=q_3=q then DL is the Cayley graph of a
finitely presented group into which the lamplighter group embeds naturally.
Also 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
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 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.

27
The categories of this paper are: comp-gas
18
The categories of this paper are: econ
0
The categories of this paper are: hep-ph
14
The categories of this paper are: math-ph
33
The categories of this paper are: plasm-ph
9
The categories of this paper are: nlin
36
The categories of this paper are: cmp-lg
3
The categories of this paper are: physics
23
The categories of this paper are: solv-int
20
The categories of this paper are: acc-phys
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The categories of this paper are: mtrl-th
7
The categories of this paper are: hep-th
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The categories of this paper are: chao-dyn
1
The categories of this paper are: cs
10
The categories of this paper are: q-bio
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The categories of this paper are: bayes-an