## 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)].

### MSC:

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

Zbl 1062.22010
Full Text:

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