Virtual compact specialness of hyperbolic one-relator groups

OpenAI

Every word-hyperbolic one-relator group is virtually compact special: a finite-index subgroup is the fundamental group of a finite nonpositively curved cube complex locally isometric to a Salvetti complex. With Kielak–Linton this yields virtual free-by-cyclic structure, allowing an infinite-rank free kernel.

Sources for this record

What was already known

Wise established virtual compact specialness for one-relator groups with torsion; Linton handled groups with free two-generator subgroups and reduced remaining hyperbolic cases to primitive extension groups. The general hyperbolic one-relator case remained open.

AI involvement AI-led

OpenAI is credited as author of this internal-model manuscript. Wise, Linton, Kielak and others supplied earlier reductions and consequences.

OpenAI is the credited author and identifies internal-model production; earlier human theorems are inputs, not the claimed new proof.

Importance Major

Claims to close the general hyperbolic one-relator virtual-specialness problem.

Limitations

The proof uses a companion Baumslag–Solitar-free hyperbolicity theorem at two points; neither that input nor the new cubulation argument has a documented completed check.

Technical verification Reported

The source supplies a long proof but neither its repository formalization catalog nor targeted follow-up search documents a completed check of this theorem; Reported is the evidence default.

Human understanding Few

No named person explains the new selective-filling and cubulation mechanism; this is an evidence default.

Correction and publication history