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.

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.