Non-stability of Liouville measures under convex combinations

TL;DR

The paper resolves a longstanding question by showing that, for every countable amenable group that is not hyper-FC-central and for every $k\ge 2$, one can construct a sequence of symmetric, non-degenerate measures such that any convex combination involving fewer than $k$ of them is Liouville, while any combination with $k$ or more is non-Liouville; in particular, the set of Liouville measures is not closed under finite convex combinations. The authors develop a sophisticated probabilistic framework based on record times, long gaps, fit times, and a Følner–switching architecture to build the measures and analyze their Poisson boundaries. They prove two principal results: a general A-type theorem for non-hyper-FC-central groups and a finite-entropy B-type extension for finitely Liouville groups, with the latter relying on the ${\mathcal D}$-metric and entropy continuity. The constructions yield both explicit non-Liouville examples and Liouville extreme points, illuminating rich boundary behavior and providing new negative answers to stability questions about Liouville properties; notable instances include lamplighter groups over $\mathbb{Z}$ and $\mathbb{Z}^2$, and the infinite symmetric group on $\mathbb{N}$.

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 $\mathbb{Z}$.

Theorems & Definitions (23)

• Theorem A: = Theorem \ref{['thm:main1']}
• Theorem B: =Theorem \ref{['thm:entropy']}
• proof
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• Theorem 2.10
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• Theorem 3.7
• proof