Wilton’s conjecture on groups with uniform negative immersions

Prove that the fundamental group of every finite 2-complex with uniform negative immersions is a locally quasi-convex hyperbolic group.

Background

Uniform negative immersions yield strong control of subgroup complexity via linear programming arguments and curvature bounds. Wilton conjectures that these combinatorial constraints imply both hyperbolicity and local quasi-convexity.

References

Wilton has conjectured that all finite 2-complexes with uniform negative immersions have locally quasi-convex hyperbolic fundamental group in Conjecture 12.9.

— The theory of one-relator groups: history and recent progress (2501.18306 - Linton et al., 30 Jan 2025) in Subsection 6.4 (Virtually compact special one-relator groups)