Poisson Boundaries of Low-Dimensional Wreath Products

Definitions and mathematical statement

Retain the source's group notation $G_d={\mathbb{Z}}^d\wr B$ for $d\in\{3,4\}$. Under the standard restricted-wreath convention, \[ {\mathbb{Z}}^d\wr B=\left(\bigoplus_{b\in B}{\mathbb{Z}}^d\right)\rtimes B, \] where $B$ shifts the indices. This makes ${\mathbb{Z}}^d$ the lamp group and $B$ the base group. Some lamplighter discussions use the opposite order, so this convention cannot be exchanged with $B\wr{\mathbb{Z}}^d$ without confirmation. The group $B$ is not defined in the extract.

For a specified finite symmetric generating set $S$ of $G_d$, simple random walk ordinarily has uniform step law $\mu$ on $S$ and transitions $g\mapsto gs$. Its bounded harmonic functions satisfy \[ h(g)=\sum_{s\in S}\mu(s)h(gs). \] To identify its Poisson boundary is to give a measured $G_d$-space $(\mathcal B,\nu)$ representing all bounded $\mu$-harmonic functions via the Poisson transform \[ h(g)=\int_{\mathcal B}f(g\xi)\,{\,\mathrm{d}}\nu(\xi),\qquad f\in L^\infty(\mathcal B,\nu), \] with the usual measure-theoretic identification. Merely producing a nontrivial boundary factor or a candidate limiting configuration is not the full identification.

The source asks for this boundary in dimensions $d=3$ and $d=4$. The answer cannot be uniquely specified without $B$, the wreath-product action/order convention and the generating set or step law. The supplied assessment discusses finite-second-moment results for free metabelian groups; those hypotheses and groups do not by themselves define the written wreath-product walk. No particular lamp-configuration boundary or metabelian replacement is asserted.

Short English statement

Identify the full Poisson boundaries of the stated simple walks for $d=3,4$. The group $B$, wreath order and walk law are unspecified, and the supplied metabelian partial results cannot be substituted for those missing definitions.

Sources

• Supplied report: Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/42, pp. 2337--2406; proposed year 2008. https://doi.org/10.4171/owr/2008/42.
• Supplied Lyons--Peres, JEMS, reference: https://ems.press/content/serial-article-files/32538.
• Supplied Erschler, Commentarii Mathematici Helvetici, reference: https://doi.org/10.4171/CMH/220.
• Local UnsolvedMath registry, permanent ID 30001030, catalogue entry; created 25 August 2026, updated 16 September 2026. Its corrected_verified note says the formula was retained with notation repaired. Literature checked at 2026-08-21: the assessment is supplied dated curation, not independently verified. It labels the record partially solved, reporting a free-metabelian boundary extension in rank at least three under finite second moment, while retaining a broader question; its relation to the exact wreath-product conventions is not specified.