Choquet-Deny groups and the infinite conjugacy class property. (English) Zbl 1428.60013

Let \(G\) be a countable discrete group. A probability measure on \(G\) is nondegenerate if its support generates \(G\) as a semigroup. \(G\) is called a Choquet-Deny group, if for every nondegenerate probability measure \(\mu\) on \(G\), all bounded \(\mu\)-harmonic functions are constant. The authors show that a finitely generated group \(G\) is Choquet-Deny, if and only if it is virtually nilpotent. For general countable discrete groups, it is shown 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, nondegenerate measure. For a partial result in the above direction see [W. Jaworski, Can. Math. Bull. 47, No. 2, 215–228 (2004; Zbl 1062.22010)].


60B15 Probability measures on groups or semigroups, Fourier transforms, factorization


Zbl 1062.22010
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