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An entropy criterion for maximality of the boundary of random walks on discrete groups. (English. Russian original) Zbl 0611.60060
Sov. Math., Dokl. 31, 193-197 (1985); translation from Dokl. Akad. Nauk SSSR 280, 1051-1054 (1985).
The following geometrical criterion for the maximality of an exit boundary of a random walk on a countable group G with distribution $$\mu$$ is given (without detailed proofs). Let $$(\Gamma_{\xi},\nu_{\xi})$$ be a measurable quotient of the tail $$\sigma$$-algebra ($$\Gamma$$,$$\nu)$$. Suppose given a metric space (X,d) together with measurable maps $$\theta$$ : $$G\to X$$ and $$\pi_ n: \Gamma_{\xi}\to X$$, $$n\in {\mathbb{N}}$$, satisfying $\log | \{g\in G: d(\theta (g),x)\leq n\}| \leq C_ n\text{ for every } x\in X$ and $$d(\theta (y_ n),\pi_ n(y_{\infty}))=O(n)$$ for $$P^{\mu}$$ a.e. trajectory $$(y_ n)^{\infty}_{n=1}$$, where $$y_{\infty}=\lim_{n\to \infty}y_ n\in \Gamma$$. Then $$(\Gamma_{\xi},\nu_{\xi})$$ is canonically isomorphic to ($$\Gamma$$,$$\nu)$$.
This criterion is deduced from the result that $$(\Gamma_{\xi},\nu_{\xi})=(\Gamma,\nu)$$ if and only if the conditional entropies h(G,$$\mu$$,$$\gamma)$$, $$\gamma\in \Gamma$$, are zero for $$\nu$$ a.a. $$\gamma\in \Gamma$$ [c.f. the author and A. M. Vershik, Random walks on discrete groups: Boundary and entropy. Ann. Probab. 11, 457-490 (1983)].
The criterion applies to the following cases, which include all known examples (note however that $$\mu$$ is restricted to be of finite entropy): (i) Discrete subgroups of semi-simple groups; (ii) fundamental groups of compact manifolds with negative sectional curvature; (iii) free products of finitely many finitely generated groups.
Note: Corrections to the translation: P. 193, line -7 should read: ”The tail $$\sigma$$-algebra coincides $$P^{\mu}$$-mod 0 with the stationary one, and thus....”. P. 196, line 8 should read: ”... a sequence of geodesics in M....”.
Reviewer: C.Series

MSC:
 60G50 Sums of independent random variables; random walks 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization 22E40 Discrete subgroups of Lie groups 28D20 Entropy and other invariants