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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
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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 free groups, one-relator groups, right-orderability
MSC Classifications: 20F65 show english descriptions Geometric group theory [See also 05C25, 20E08, 57Mxx] 20F65 - Geometric group theory [See also 05C25, 20E08, 57Mxx]
 

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