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 - 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 - 10.4007/annals.2019.190.1.5
DO - 10.4007/annals.2019.190.1.5
M3 - Article
AN - SCOPUS:85069454818
SN - 0003-486X
VL - 190
SP - 307
EP - 320
JO - Annals of Mathematics
JF - Annals of Mathematics
IS - 1
ER -