Geodesic
Group representation of the Cayley forest and some of its applications
Izvestiya. Mathematics , Tome 67 (2003) no. 1, pp. 17-27

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Cayley forests and products of Cayley trees of order $k\geqslant 1$ are represented as subgroups in the free product of $m$ cyclic groups ($m>k$) of order 2. The automorphism groups of these objects are determined. We give a complete description of the sets of translation-invariant and periodic Gibbs measures for the Ising model on a Cayley forest. We construct a new class of limiting Gibbs measures for the inhomogeneous Ising model on a Cayley tree. We find sufficient conditions for random walks in periodic random media on the forest to be never returning provided that the jumps of the walking particle are bounded. 
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     title = {Group representation of the {Cayley} forest and some of its applications},
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N. N. Ganikhodzhaev; U. A. Rozikov. Group representation of the Cayley forest and some of its applications. Izvestiya. Mathematics , Tome 67 (2003) no. 1, pp. 17-27. http://geodesic.mathdoc.fr/item/IM2_2003_67_1_a1/