Følner Growth and Trivial Poisson Boundary

Definitions and mathematical statement

Fix a finite symmetric generating set $S$ for a finitely generated group $G$. In a standard vertex-boundary convention, put \[ \partial_S F=FS\setminus F,\qquad \operatorname{Fol}_{G,S}(n)=\min\{|F|:\varnothing\ne F\subset G\text{ finite},\ |\partial_SF|\leq |F|/n\}, \] with value $\infty$ if there is no such $F$. The source does not specify the boundary convention or the usual comparison equivalence for Følner functions.

For simple random walk associated with $S$, write $\mu_S$ for the uniform distribution on $S$. Triviality of its Poisson boundary is equivalently the Liouville property: every bounded $\mu_S$-harmonic function is constant, where \[ f(g)=\frac1{|S|}\sum_{s\in S}f(gs)\quad(g\in G). \] The source asks whether the implication \[ \operatorname{Fol}_{G}(n)\leq e^n \quad\Longrightarrow\quad \text{the simple random walk on }G\text{ has trivial Poisson boundary} \] holds for finitely generated groups. In the displayed fixed-$S$ interpretation this means the stated bound on $\operatorname{Fol}_{G,S}$ implies the Liouville property for $\mu_S$. The inequality is retained as written, not replaced by a subexponential bound or an unspecified $e^{Cn}$ estimate. Its exact range in $n$, eventual versus all-$n$ meaning, and possible coarse rescaling need source confirmation. Finiteness of the Følner function at arbitrarily large scales already places the group in the amenable setting; no separate restriction to infinite groups is supplied.

The generating set for the Følner function and the law of the "simple" walk must be fixed consistently. The record does not say whether the premise is meant for some generating set, every generating set, or a coarse equivalence class, nor whether the conclusion is for one or all finite generating-set walks. It does not ask here about arbitrary nonsymmetric or infinite-support measures.

The supplied triage checked 2026-08-21 labels the question open, saying no known characterization yields boundary triviality from this bound and describing the amenable-group problem as unresolved. These are dated attributed assessments.

Short English statement

Ask whether the stated exponential Følner upper bound forces every bounded harmonic function of the associated simple random walk to be constant. The generating set, Følner normalization and exact meaning of the bound require source confirmation.

Sources

• Local UnsolvedMath registry, permanent ID 30001035, source code catalogue entry; supplied complete source record and corrected-verified notation-cleanup note. Literature checked 2026-08-21: the assessment is supplied dated curation, not independently verified here.
• Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/42, pp. 2337--2406: https://doi.org/10.4171/owr/2008/42. Proposed-year metadata gives 2008.
• Erschler--Frisch, Poisson boundaries: https://arxiv.org/abs/2206.11111, as labeled in the supplied record.