Poisson–Furstenberg boundary and growth of groups

Bartholdi, Laurent; Erschler, Anna

doi:10.1007/s00440-016-0712-6

Cited by 13 publications

(11 citation statements)

References 25 publications

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“…Furthermore, by taking the group morphism of H 1 into Z ≀ H 1 , we see that the image of h 1 is the generator p1, 0q of the active group, while for every j, the image of h 1 1 j is of the form p0, f j q where f j has finite support. The following result is essentially due to Kaimanovich and Vershik [26, Proposition 6.1], [21,Theorem 1.3], and has been studied in a more general context by Bartholdi and Erschler [6]: Lemma 9.2. Consider the wreath product Z ≀ H 1 where H 1 is not trivial, and let µ be a measure on it such that the projection of µ on Z gives a transient walk and the projection of µ on H 1 Z is finitary and non-trivial.…”

Section: An Algebraic Lemma and Proof Of The Main Resultsmentioning

confidence: 99%

Non-triviality of the Poisson boundary of random walks on the group of Monod

Bogdan¹

2019

Ergod. Th. Dynam. Sys.

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We give sufficient conditions for the non-triviality of the Poisson boundary of random walks on HpZq and its subgroups. The group HpZq is the group of piecewise projective homeomorphisms over the integers defined by Monod. For a finitely generated subgroup H of HpZq, we prove that either H is solvable, or every measure on H with finite first moment that generates it as a semigroup has non-trivial Poisson boundary. In particular, we prove the non-triviality of the Poisson boundary of measures on Thompson's group F that generate it as a semigroup and have finite first moment, which answers a question by Kaimanovich.

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Section: An Algebraic Lemma and Proof Of The Main Resultsmentioning

confidence: 99%

Non-triviality of the Poisson boundary of random walks on the group of Monod

Bogdan¹

2019

Ergod. Th. Dynam. Sys.

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“…The inverted orbit is a well-known object, central to the study of growth and random walks on permutational wreath products, see [BE12,BE11,AV12]. It is sometimes convenient to consider the following variation of the inverted orbit…”

Section: Probabilistic Reformulation and Recurrent Actions 41 The Inmentioning

confidence: 99%

Extensive amenability and an application to interval exchanges

Juschenko¹,

Bon

Monod

2016

Ergod. Th. Dynam. Sys.

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Extensive amenability is a property of group actions which has recently been used as a tool to prove amenability of groups. We study this property and prove that it is preserved under a very general construction of semidirect products. As an application, we establish the amenability of all subgroups of the group IET of interval exchange transformations that have angular components of rational rank ≤ 2.In addition, we obtain a reformulation of extensive amenability in terms of inverted orbits and use it to present a purely probabilistic proof that recurrent actions are extensively amenable. Finally, we study the triviality of the Poisson boundary for random walks on IET and show that there are subgroups G < IET admitting no finitely supported measure with trivial boundary.

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“…Kaimanovich and Vershik [21, p. 466] conjecture that "Given an exponential group G, there exists a symmetric (nonfinitary, in general) measure with non-trivial boundary." See Bartholdi and Erschler [2] for additional related results and further references and discussion.…”

Section: Introductionmentioning

confidence: 98%

Choquet-Deny groups and the infinite conjugacy class property

et al. 2019

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A countable discrete group G is called Choquet-Deny if for every non-degenerate probability measure µ on G it holds that all bounded µ-harmonic functions are constant. We show that a finitely generated group G is Choquet-Deny if and only if it is virtually nilpotent. For general countable discrete groups, we show that G is Choquet-Deny if and only if none of its quotients has the infinite conjugacy class property. Moreover, when G is not Choquet-Deny, then this is witnessed by a symmetric, finite entropy, non-degenerate measure.

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Poisson–Furstenberg boundary and growth of groups

Cited by 13 publications

References 25 publications

Non-triviality of the Poisson boundary of random walks on the group of Monod

Non-triviality of the Poisson boundary of random walks on the group of Monod

Extensive amenability and an application to interval exchanges

Choquet-Deny groups and the infinite conjugacy class property

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