Choquet-Deny groups and the infinite conjugacy class property

Joshua Frisch; Yair Hartman; Omer Tamuz; Pooya Vahidi Ferdowsi

doi:https://doi.org/10.4007/annals.2019.190.1.5

Choquet-Deny groups and the infinite conjugacy class property

Joshua Frisch, Yair Hartman, Omer Tamuz, Pooya Vahidi Ferdowsi

Department of Mathematics

Ben-Gurion University of the Negev

Research output: Contribution to journal › Article › peer-review

Abstract

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.

Original language	English
Pages (from-to)	307-320
Number of pages	14
Journal	Annals of Mathematics
Volume	190
Issue number	1
DOIs	https://doi.org/10.4007/annals.2019.190.1.5
State	Published - 1 Jul 2019

Keywords

Furstenberg-Poisson boundary
Harmonic functions
Random walks

All Science Journal Classification (ASJC) codes

Statistics and Probability
Statistics, Probability and Uncertainty

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https://doi.org/10.4007/annals.2019.190.1.5

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title = "Choquet-Deny groups and the infinite conjugacy class property",

abstract = "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.",

keywords = "Furstenberg-Poisson boundary, Harmonic functions, Random walks",

author = "Joshua Frisch and Yair Hartman and Omer Tamuz and Ferdowsi, {Pooya Vahidi}",

note = "Funding Information: Keywords: Furstenberg-Poisson boundary, random walks, harmonic functions AMS Classification: Primary: 60B15. J. Frisch was supported by NSF Grant DMS-1464475. Y. Hartman was partially supported by the Israel Science Foundation (grant No. 1175/18). He is grateful for the support of Northwestern University, where he was a postdoctoral fellow when most of this research was conducted. O. Tamuz was supported by a grant from the Simons Foundation (#419427). {\textcopyright}c 2019 Department of Mathematics, Princeton University. 1In the context of Markov chains such measures are called irreducible. Publisher Copyright: {\textcopyright} 2019 Department of Mathematics, Princeton University.",

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AU - Ferdowsi, Pooya Vahidi

N1 - Funding Information: Keywords: Furstenberg-Poisson boundary, random walks, harmonic functions AMS Classification: Primary: 60B15. J. Frisch was supported by NSF Grant DMS-1464475. Y. Hartman was partially supported by the Israel Science Foundation (grant No. 1175/18). He is grateful for the support of Northwestern University, where he was a postdoctoral fellow when most of this research was conducted. O. Tamuz was supported by a grant from the Simons Foundation (#419427). ©c 2019 Department of Mathematics, Princeton University. 1In the context of Markov chains such measures are called irreducible. Publisher Copyright: © 2019 Department of Mathematics, Princeton University.

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N2 - 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.

AB - 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.

KW - Furstenberg-Poisson boundary

KW - Harmonic functions

KW - Random walks

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JO - Annals of Mathematics

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ER -

Choquet-Deny groups and the infinite conjugacy class property

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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