Non-stability of Liouville measures under convex combinations

Abstract: For every non-hyper-FC-central countable amenable group and every $k\geq 2$, we provide a sequence of symmetric, fully supported probability measures such that their convex combination is non-Liouville (that is it admits a non-constant bounded harmonic function, equivalently, the Poisson boundary is non-trivial) if and only if at least $k$ of them appear in the convex combination. Particularly, our result implies that the set of Liouville measures is not closed under convex combination, which answers a question of Kaimanovich. We also provide a similar result under the additional assumption of finite entropy for those non-hyper-FC-central countable groups with the property that every symmetric, finitely supported probability measure is Liouville. These groups are the only known non-trivial examples of countable groups that admit Liouville measures with finite entropy. Examples include the lamplighter group over $\mathbb{Z}$ and $\mathbb{Z}^2$, and the infinite symmetric group of finite permutations on $\mathbb{Z}$.