Return Probabilities of Amenable Groups with Exponential Growth

Definitions and mathematical statement

Let $G$ be a finitely generated group with identity $e$. For a finite symmetric generating set $S$, let $B_S(n)$ be the set of elements representable by words of length at most $n$ in $S$. The group has exponential growth if there exist $c>1$ and $n_0\in{\mathbb{N}}$ such that $|B_S(n)|\ge c^n$ for every integer $n\ge n_0$; this property is independent of the finite generating set. The group is amenable if there is a finitely additive function $m:\mathcal P(G)\to[0,1]$ with $m(G)=1$ and $m(gA)=m(A)$ for all $g\in G$ and $A\subseteq G$.

The supplied question asks for a characterization of the finitely generated amenable groups $G$ of exponential growth satisfying \[ p^G_{2n}\asymp\exp(-n^{1/3})\qquad(n\to\infty). \] The notation requires qualifications absent from the supplied record. For a probability measure $\mu$ on $G$, independent increments $Y_1,Y_2,\ldots$ with law $\mu$ define a random walk $X_0=e$, $X_k=Y_1\cdots Y_k$, and its return probability is \[ p^{G,\mu}_{2n}:=\mathbb P(X_{2n}=e)=\mu^{*2n}(e). \] The source does not specify which measure defines $p^G_{2n}$, whether symmetry, finite support, or generating support is required, or whether the desired condition concerns one chosen measure, some admissible measure, or every admissible measure. These choices remain unresolved.

The comparison convention is also unspecified. Multiplicative comparability would mean that there exist $C_1,C_2>0$ and $n_0\in{\mathbb{N}}$ such that \[ C_1\exp(-n^{1/3})\le p^G_{2n}\le C_2\exp(-n^{1/3}) \quad\text{for every integer }n\ge n_0. \] A comparison allowing constant rescaling of time would instead permit bounds of the form \[ \exp(-b n^{1/3})\le p^G_{2n}\le\exp(-a n^{1/3}) \quad\text{for every integer }n\ge n_0, \] for some $0<a\le b<\infty$. These are different requirements. The supplied evidence does not determine which is intended, so the displayed source question is retained with these unresolved qualifications rather than assigned a definitive interpretation.

Short English statement

Characterize the finitely generated amenable groups of exponential growth whose even-time return probabilities have the supplied decay $p^G_{2n}\asymp\exp(-n^{1/3})$. The random-walk measure, its quantifier, and the comparison convention require clarification. The supplied dated curation labels the problem partially solved and reports a characterization for finitely generated metabelian groups in terms of Krull dimension, while describing the broader question as unresolved; these claims have not been independently verified.

Sources

• Local UnsolvedMath registry, permanent ID $30003211$, source problem code catalogue entry, title: Return Probabilities of Amenable Groups with Exponential Growth. No individual proposer is supplied; the recorded proposed year is 2016.
• Supplied report citation: Measured Group Theory (2017), Oberwolfach Reports, DOI 10.4171/owr/2016/41, pp. 2347--2397. Original-question link: https://doi.org/10.4171/owr/2016/41. The background attributes the question to workshop open-problem discussions documented in Oberwolfach Reports.
• Supplied literature reference, labeled "Jacoboni, metabelian groups": https://numdam.org/articles/10.5802/aif.3291/. No theorem statement or full bibliographic details are supplied.
• Literature assessment checked at 2026-08-21: supplied dated curation, not independently verified. It labels the problem partially solved, reports a characterization for finitely generated metabelian groups in terms of Krull dimension, and states that the question for all finitely generated amenable groups of exponential growth remains unresolved beyond that class. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review asserting that the clean statement is self-contained and preserves the target. This is supplied review metadata, not independent verification of the unresolved conventions above.