Gersten’s conjecture and virtual compact specialness of one-relator groups

Proves Gersten's conjecture: every finitely generated one-relator group containing no Baumslag–Solitar subgroup \mathrm{BS}(m,n), with m,n\ne0, is word-hyperbolic. It also proves that every word-hyperbolic one-relator group is virtually compact special.

Manuscripts (2)

Baumslag-Solitar-free one-relator groups are hyperbolic

September 25, 2026 · source · cite as \cite{OAI:Baumslag-Solitar-free-one-relator-groups-are-hyperbolic-September-25-2026}

We prove Gersten's conjecture: every finitely generated one-relator group containing no subgroup isomorphic to a Baumslag–Solitar group \mathop{\mathrm{BS}}\nolimits (m,n), for nonzero integers m, n, is word-hyperbolic.

Virtual compact specialness of hyperbolic one-relator groups

September 25, 2026 · source · cite as \cite{OAI:Virtual-compact-specialness-of-hyperbolic-one-relator-groups-September-25-2026}

We prove that every word-hyperbolic one-relator group is virtually compact special. Combined with the work of Kielak–Linton, this resolves Wise's virtual free-by-cyclic conjecture for hyperbolic one-relator groups: every such group is virtually free-by-cyclic, with the free kernel allowed to have infinite rank.

Result 258 · Group theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.