TY - JOUR

T1 - Choquet-Deny groups and the infinite conjugacy class property

AU - Frisch, Joshua

AU - Hartman, Yair

AU - Tamuz, Omer

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.

PY - 2019/7/1

Y1 - 2019/7/1

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

UR - http://www.scopus.com/inward/record.url?scp=85069454818&partnerID=8YFLogxK

U2 - https://doi.org/10.4007/annals.2019.190.1.5

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

M3 - Article

SN - 0003-486X

VL - 190

SP - 307

EP - 320

JO - Annals of Mathematics

JF - Annals of Mathematics

IS - 1

ER -