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.

Non-triviality of the Poisson boundary of random walks on the group of Monod

“…This lemma, along with Lemma 3.2.4 implies: We will now show how measures satisfying whose assumptions can be constructed. Remark that the question of existence of a measure with non-trivial boundary has already been solved by Frisch-Hartman-Tamuz-Vahidi-Ferdowski [16]. In our case, notice that A Ă H s (see (1)), and it is isomorphic to Z.…”

“…If the subgroup is also non-abelian, we have proven that it contains a wreath product of Z with another subgroup (see (10)). In particular, it is not virtually nilpotent, which implies (as it is finitely generated) that there exists a measure on it with non-trivial boundary by a recent result of Frisch-Hartman-Tamuz-Vahidi-Ferdowski [16]. Furthermore, it is known that on the wreath products Z ≀ Z it is possible to obtain a measure with finite 1´ε moment and non-trivial Poisson boundary for every ε ą 0 (see Lemma 9.2 and discussion before and after it).…”

“…It is also known that amenable groups can have measures with non-trivial boundary. In a recent result Frisch-Hartman-Tamuz-Vahidi-Ferdowski [16] describe an algebraic necessary and sufficient condition for a group to admit a measure with non-trivial boundary. In the present paper we give sufficient conditions for nontriviality of the Poisson boundary on HpZq.…”

Isoperimetric profiles and random walks on some groups defined by piecewise actions

“…Further references are found in the listed articles. Some recent results on harmonic functions and group structure are in [2,4,5,7,19,21,22,26]. Entropy and/or displacement are discussed in [1,6,12,16,27,29].…”

Examples of measures with trivial left and non-trivial right random walk tail boundary

“…It is well known that amenable groups and only them admit non-degenerate Liouville measures, see Theorems 4.2 and 4.3 from [KaVe83]). It is also well known that all measures on groups without ICC factors are Liouville, see [Ja], a self-contained proof could be also found in the second preprint version of [Feta19]. Thus examples of Kaimanovich type are possible only for amenable groups with non-trivial ICC factors.…”