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# Stackings and the $W$-cycles Conjecture

Published:2017-04-13
Printed: Sep 2017
• Larsen Louder,
University College London, London, UK
• Henry Wilton,
University of Cambridge, Cambridge, UK
 Format: LaTeX MathJax PDF

## 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 finitely generated Schur multiplier.
 Keywords: free groups, one-relator groups, right-orderability
 MSC Classifications: 20F65 - Geometric group theory [See also 05C25, 20E08, 57Mxx]

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