Generating Sets Realizing Minimal Growth

Definitions and mathematical statement

Let $F_n$ be a free group of finite rank $n$, and let $\phi\in\operatorname{Aut}(F_n)$. Write \[ G=F_n\rtimes_\phi{\mathbb{Z}}. \] The supplied record does not specify the intended range of $n$ or impose further conditions on $\phi$.

For a finite generating set $X\subseteq G$, put $S_X=X\cup X^{-1}$ and define \[ |g|_X=\min\{k\geq 0:g=s_1\cdots s_k,\ s_i\in S_X\}, \qquad B_X(r)=\{g\in G:|g|_X\leq r\}. \] Use the standard multiplicative exponential growth convention \[ e(G,X)=\lim_{r\to\infty}|B_X(r)|^{1/r}, \qquad r\in{\mathbb{Z}}_{\geq1}. \] This limit exists by submultiplicativity of ball cardinalities. The supplied record does not itself define $e(G,X)$; the convention above makes the draft explicit and requires confirmation against the report. Using its logarithm instead would give the same minimizing generating sets.

Define the growth spectrum \[ \xi(G)=\{e(G,X):X\subseteq G\text{ is finite and }\langle X\rangle=G\}. \] Problem. Characterize all finite generating sets $X$ of $G$ satisfying \[ e(G,X)=\min\xi(G). \] This formulation presupposes that $\xi(G)$ has a minimum. The supplied background reports attainment in relevant groups but does not state the precise hypotheses or a theorem establishing attainment throughout the intended free-by-cyclic class. Thus the requested characterization is stated subject to this unresolved qualification; attainment for the full intended class is not asserted here.

Short English statement

For a free-by-cyclic group, characterize every finite generating set whose exponential growth rate is smallest among all finite generating sets. The intended rank range, growth convention, and hypotheses ensuring that the minimum is attained require confirmation. This is a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006192$, title Generating Sets Realizing Minimal Growth, exact source code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied proposed year is 2025; no individual proposer is identified.
• Geometric Structures in Group Theory (2025), Oberwolfach Reports, pp. 275--338, DOI 10.4171/owr/2025/6: https://doi.org/10.4171/owr/2025/6. The record describes these reports as documenting mathematical workshops and their open-problem discussions. Its original-statement field contains only a contextual fragment referring to Problem 4, rather than a complete mathematical statement.
• Supplied literature source: OWR 2025/6 geometric group theory report, https://oa.tib.eu/renate/server/api/core/bitstreams/6f61d540-8e50-41f3-97a8-6e8214672460/content.
• Supplied literature source: Fujiwara and Sela, rates of growth, https://doi.org/10.1007/s00222-023-01200-w.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, reports known existence and finiteness properties of growth spectra, including attainment of a minimum in relevant groups, and says that no characterization of all minimizing finite generating sets for free-by-cyclic groups was found. This is supplied dated curation derived from the OWR material, not an independently verified assessment; time-sensitive status and cited claims require independent checking.
• The supplied statement status is corrected_verified, with a supplied assertion that the statement was reconstructed and checked against the official Oberwolfach report. That assertion is recorded as provenance and is not an independent verification by this draft.