Group representation of the Cayley forest and some of its applications

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.