Virtual specialness of hyperbolic one-relator groups
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Is every word-hyperbolic one-relator group virtually compact special? This means a finite-index subgroup is the fundamental group of a finite nonpositively curved cube complex locally isometric to a Salvetti complex.
Why it matters and what counts as progress
Why it matters: It asks for a strong cubical structure throughout the hyperbolic one-relator class. The related virtual free-by-cyclic consequence may have an infinite-rank free kernel.
What counts as progress: Establish virtual compact specialness for the unrestricted hyperbolic one-relator class; retain the distinction between finite and infinite rank free kernels in consequences.
How the problem developed
- 2021 · Partial result [1]
Daniel T. Wise Proved virtual compact specialness for one-relator groups with torsion, a subclass of hyperbolic one-relator groups.
- 25 Sept 2026 · Reported claim [2]
OpenAI Reported virtual compact specialness for every word-hyperbolic one-relator group; together with the cited L²-Betti-number argument this yields virtual free-by-cyclic structure, possibly with an infinite-rank free kernel.
AI contribution
Virtual compact specialness of hyperbolic one-relator groups
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
Website timeline
Publication dates and editorial changes.
OpenAI reports virtual specialness of hyperbolic one-relator groups