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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

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