Stackings and the W-cycles conjecture

Abstract

We prove Wise’s W-cycles conjecture: Consider a compact graph Γ′ immersing into another graph Γ. For any immersed cycle Λ : S¹ → Γ, we consider the map Λ′ from the circular components S of the pullback to Γ′. Unless Λ′ is reducible, the degree of the covering map S → S¹ is bounded above by minus the Euler characteristic of Γ′. As a corollary, any finitely generated subgroup of a one-relator group has finitely generated Schur multiplier.