TAP Infinity for Free-by-Cyclic Groups

Definitions and mathematical statement

A finitely generated free group is a group isomorphic to the free group $F_n$ on a finite basis of size $n$. The original source describes the groups under consideration as $\{\text{finitely generated free}\}$-by-cyclic groups. In extension notation, this means groups $G$ admitting an exact sequence \[ 1\longrightarrow F_n\longrightarrow G\longrightarrow C\longrightarrow 1, \] where $F_n$ is finitely generated and free and $C$ is cyclic. The supplied excerpt does not specify whether "cyclic" means infinite cyclic or also permits finite cyclic quotients; the intended range of $C$ must therefore be checked before this becomes a fully specified quantified statement.

Conjecture 6 (as supplied). Does every $\{\text{finitely generated free}\}$-by-cyclic group $G$ belong to \[ \operatorname{TAP}_{\infty}(R)? \] The supplied record does not define $\operatorname{TAP}_{\infty}(R)$, specify the nature or hypotheses of the parameter $R$, or state its quantifier. Membership in this class is consequently retained as source notation, without an invented definition. The question is a statement schema pending recovery of those definitions and conventions. In particular, the original wording requires the free normal subgroup to be finitely generated; this requirement must not be replaced merely by finite generation of $G$.

Short English statement

The supplied Conjecture 6 asks whether every group with a finitely generated free normal subgroup and cyclic quotient lies in $\operatorname{TAP}_{\infty}(R)$. A complete formulation requires the source definition of this class, the assumptions and quantifier on $R$, and the convention for cyclic quotients. The registry labels the problem open; this is supplied curation, not an independently verified status.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005699$, title "TAP Infinity for Free-by-Cyclic Groups," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified in the supplied record.
• Supplied source citation: Mini-Workshop: Homological aspects of TDLC-groups (2024), Oberwolfach Reports, pp. 3103--3138, DOI 10.4171/owr/2023/55: https://doi.org/10.4171/owr/2023/55. The original excerpt identifies Conjecture 6 on p. 3127. The registry gives proposed year 2023; the supplied citation gives 2024.
• Supplied literature source, "Oberwolfach report": https://ems.press/content/serial-article-files/48166.
• Supplied literature source, "Hughes--Kudlinska": https://doi.org/10.1112/plms.70059.
• Supplied literature source, "Profinite rigidity of fibring": https://arxiv.org/abs/2206.11347.
• Literature assessment dated exactly 2026-08-22: supplied dated curation, not independently verified. It labels the problem open and states that known TAP results cover LERF and products of limit groups, while the 2025 free-by-cyclic work does not prove TAP infinity for every finitely generated free-by-cyclic group. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that it was checked against the official Oberwolfach report. This is a supplied verification claim; no source consultation or independent mathematical verification is asserted here.