# Can a lacunary hyperbolic group be Liouville?

While discussing with a colleague of a possible use of lacunary hyperbolic elementarily amenable groups introduced by Olshanskii, Osin & Sapir, it occurred to me that I was not aware whether these groups (or any amenable lacunary hyperbolic group) could be Liouville (in the sense that there are no non-constant harmonic functions on some Cayley graph). Hence the question in the title:

Question: Are there lacunary hyperbolic groups which admit a Cayley graph so that there is no non-constant harmonic function on that graph?

Or more restrictively, can this be said of the groups introduced by Olshanskii, Osin & Sapir (Lacunary hyperbolic groups, section 3.5)?

• Just for context, being Liouville is an analytic/probabilistic strengthening of being amenable, which is neither weaker nor stronger than being elementary amenable / virtually solvable, and satisfied by both virtually polycyclic groups and subexponential growth groups. I unfortunately found no survey. I think there are f.g. solvable non-Liouville groups, possibly metabelian (are Liouville groups classified among metabelian groups?)
– YCor
Apr 29 at 7:48
• @Ycor (1) the free metabelian group with two generators is Liouville [Erschler?] (2) if one considers harmonic functions w.r.t. to a finite supported measures (i.e. "put a weight on the edges"), then it's not too hard to see that: if a H is a quotient of G and H is not for some measure $p$ Liouville then G is not Liouville for some measure $p'$ (3) from this [and other results on which groups are not Liouville] one can see that the free metabelian groups on 3 generators is not Liouville (4) it's an open problem if being Liouville depends on the choice of generating set (or fin. supp. measure)
– ARG
Apr 29 at 13:50
• ... (5) I think it's a result of Kaimanovich-Vershik that any amenable group has a measure $p$ for which it is Liouville, but this measure is most of the time not finitely supported. (6) a metabelian group which is not Liouville is the lamplighter on $\mathbb{Z}^3$.
– ARG
Apr 29 at 13:54
• @Ycor there is a larger class than polycyclic groups which is known to be Liouville: virtually solvable minimax groups. This follows from an estimate of Kropholler and Lorensen (on the return probability) and a result of Saloff-Coste and Zheng (groups where the return probability of the random walks decays more slowly than $\mathrm{exp}(-n^{1/2})$ are Liouville).
– ARG
Apr 29 at 15:21