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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.

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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)