![]() The key observation is to deduce from this that the cocycle k g n at time n is with overwhelming probability constant on a given cylinder subset of Σ, provided that the number of letters defining the cylinder is big enough compared to √ n. This fact, using some general Gaussian estimates due to Hebisch and Saloff-Coste, implies that the maximal displacement up to time n of the orbit cocycle k g n ( x ) has typical size √ n and the tail of its distribution admits a Gaussian upper bound. Thus, if g n is the left random walk on G, k g n ( x ) performs a random walk on a graph with vertex set Z and edges connecting integers with bounded difference. This embedding is given by g x ↦ k g ( x ), where k g is the orbit cocycle. Given any subshift ( Σ, τ ), and a finitely generated subgroup G = ⟨ S ⟩ of ], the orbital Schreier graph of any non-periodic point x ∈ Σ admits a natural Lipschitz embedding into Z. Let us give an outline of the proof of Theorem 1.2. For a recent survey on Poisson-Furstenberg boundaries of random walks on discrete groups, see. However, there are finitely generated amenable groups, such as the wreath product Z / 2 Z ≀ Z 3, that admit no finitely supported, non-degenerate measures with trivial boundary, see Kaimanovich and Vershik on some amenable groups, a non-degenerate measure with trivial boundary might not even be chosen to have finite entropy by a result of Erschler. More precisely, a group is amenable if, and only if, it admits a symmetric non-degenerate measure μ with trivial Poisson-Furstenberg boundary (one implication is due to Furstenberg, see, the other to Kaimanovich and Vershik and to Rosenblatt ). Finitely generated groups with sub-exponential growth have the Liouville property (this is due to Avez ), and groups with the Liouville property are amenable. When no measure is specified, we say that the group G has the Liouville property if ( G, μ ) has the Liouville property for every symmetric, finitely supported probability measure μ on G, including degenerate measures.
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