Liouville property under rough isometry to nonamenable Cayley graphs

Definitions and mathematical statement

Let $\Gamma$ be a finitely generated group with a finite symmetric generating set $S$, and let $H=\operatorname{Cay}(\Gamma,S)$ be its connected Cayley graph. Suppose $H$ is nonamenable, meaning \[ \inf_{0<|A|<\infty}\frac{|\partial_H A|}{|A|}>0, \] where $\partial_H A$ is the external vertex boundary.

Let $G$ be a connected graph of uniformly bounded degree. Assume there is a map $F:V(G)\to V(H)$ and constants $A\ge1$, $B\ge0$ such that \[ A^{-1}d_G(x,y)-B\le d_H(F(x),F(y))\le A d_G(x,y)+B \] for all $x,y$, and every vertex of $H$ is within distance $B$ of $F(V(G))$. Thus $G$ and $H$ are quasi-isometric, termed roughly isometric in the source.

Must $G$ support a bounded nonconstant harmonic function for its own simple random walk? Explicitly, is there a real function $f$, not constant on $V(G)$, with $\sup_v|f(v)|<\infty$ and \[ f(v)=\frac1{\deg_G(v)}\sum_{w\sim_Gv}f(w)\qquad(v\in V(G))? \] The walk uses the actual unweighted edges of $G$, not transition probabilities transferred through $F$. The graph $G$ need not be transitive or itself a Cayley graph. The problem asks whether being coarsely equivalent to a nonamenable Cayley graph forces the non-Liouville property despite changes in local geometry. It is narrower than claiming quasi-isometry invariance of the Liouville property for all bounded-degree graphs.

Short English statement

Does every bounded-degree graph coarsely equivalent to a nonamenable Cayley graph admit a nonconstant bounded harmonic function?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 10000048 (AMR-099-0048), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Benjamini - Coarse Geometry and Randomness (Saint-Flour notes), Conjecture 13.2, PDF page 107. Original source indexed by AMR-099-0048. https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=107.
• S3 I. Benjamini, Coarse Geometry and Randomness, primary Saint-Flour notes; the numbered question and its surrounding definitions. https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf.