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Gersten's conjecture: are Baumslag-Solitar-free one-relator groups hyperbolic?

Theorem 1.1: if a one-relator group $\langle X\mid r\rangle$ , $X$ finite, contains no $\mathrm{BS}(m,n)$ with $m,n\ne0$ , it is word-hyperbolic. The technical core is a combination theorem for a class of HNN extensions of torsion-free hyperbolic groups with free, possibly proper edge groups, applied along the Magnus-Moldavanskii hierarchy. Combined with the companion and with Kielak-Linton, Corollary: every Baumslag-Solitar-free one-relator group is virtually free-by-cyclic, with a finite-index embedding in a free-by- $\mathbb Z$ group with finitely generated kernel. The paper does not treat infinitely generated or multi-relator groups. Unrelated to the K-theoretic Gersten conjecture addressed in family 209 of the same release.

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Occurred: Sep 25, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Gersten's conjecture: are Baumslag-Solitar-free one-relator groups hyperbolic? · Gersten's one-relator conjecture

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VibeMathed record: Gersten's conjecture: are Baumslag-Solitar-free one-relator groups hyperbolic?

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript was read against Gersten's question as the paper quotes it from Gersten (1992, p. 228); it is the conjecture in full, for every finite generating set and relator, including trivial relators and proper powers, with no torsion-freeness hypothesis. The paper notes that excluding all $\mathrm{BS}(1,n)$ is equivalent. The proof rests on Linton's one-relator hierarchy and quasiconvexity theorems and a new combination theorem for HNN extensions; it was not refereed here. There is no Lean formalization for this family (no lean/docs/258.md at the pinned commit).