An Abramov formula for stationary spaces of discrete groups

Abstract

Let (G,mu) be a discrete group equipped with a generating probability measure, and let Gamma be a finite index subgroup of G. A mu-random walk on G, starting from the identity, returns to Gamma with probability one. Let theta be the hitting measure, or the distribution of the position in which the random walk first hits Gamma. We prove that the Furstenberg entropy of a (G,mu)-stationary space, with respect to the induced action of (Gamma,theta), is equal to the Furstenberg entropy with respect to the action of (G,mu), times the index of Gamma in G. The index is shown to be equal to the expected return time to Gamma. As a corollary, when applied to the Furstenberg-Poisson boundary of (G,mu), we prove that the random walk entropy of (Gamma,theta) is equal to the random walk entropy of (G,mu), times the index of Gamma in G.Comment: 16 page

Furstenberg entropy realizations for virtually free groups and lamplighter groups

“…It is shown in [11] that E [τ ] = [G : Γ], and that for any (G, µ)-stationary space (X, ν) it holds that h θ (X, ν) = [G : Γ] · h µ (X, ν).…”

“…We also realize entropies using Bowen spaces: to prove Theorem 1, we construct Bowen spaces of lamplighter groups, and lift them to Bowen spaces of virtually free groups. Using a recent result of Hartman, Lima and Tamuz [11] (see also [13]) that relates the entropies of the actions of groups and their finite index subgroups, we control the entropies of the lifted spaces, and show that they can be made arbitrarily small.…”

“…, Z n ] ≤ C 1 + E [|L n+1 − L n ||Z n ] .By the triangle inequality and the symmetry of S it follows that= C 1 + g∈G µ(g) ||Z n g| − |Z n || ≤ C 1 + g∈G µ(g)|g|Proof of Lemma 4.4. In general, the index of Γ in G is equal to the expected hitting time[11], and so E [τ ] = [G : Γ] < ∞. It follows by Theorem (7.5) in[5] that because M n is a submartingale satisfying the condition of Claim B.1, thenE [M τ ] ≥ E [M 1 ] = 0.Hence E [τ C 1 − L τ ] ≥ 0 and since E [τ ] = [G : Γ] then E [L τ ] ≤ [G : Γ]C 1 < ∞.…”

Asymptotic entropy of transformed random walks

“…(j) Convex combinations of convolutions of a given probability measure [Kai83]. (jj) The induced random walk on a recurrent subgroup [Fur71,Kai91,Kai92,HLT14].…”

“…Although his setup was somewhat different, by the same approach one can also find the asymptotic entropy of the induced random walk on a recurrent subgroup [Kai91]. Recently, Hartman et al [HLT14] calculated the asymptotic entropy of the random walk induced on a finite index subgroup in an alternative way by using formula (iii) (although, apparently, they were not aware of [Kai92,Kai91]).…”