Stackings and the W-cycles Conjecture. Issue 3 (1st September 2017) - Louder, Larsen
Abstract: We prove Wise's $W$ -cycles conjecture. Consider a compact graph $\Gamma '$ immersing into another graph $\Gamma $ . For any immersed cycle $\Lambda :{{S}^{1}}\to \Gamma $, we consider the map $\Lambda '$ from the circular components $\mathbb{S}$ of the pullback to $\Gamma '$ . Unless $\Lambda '$ is reducible, the degree of the covering map $\mathbb{S}\to {{S}^{1}}$ is bounded above by minus the Euler characteristic of $\Gamma '$ . As a corollary, any finitely generated subgroup of a one-relator group has a finitely generated Schur multiplier.