Geodesic
Stackings and the W-cycles Conjecture
Canadian mathematical bulletin, Tome 60 (2017) no. 3, pp. 604-612

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DOI

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.
DOI : 10.4153/CMB-2016-078-3
Mots-clés : 20F65, free groups, one-relator groups, right-orderability 
Louder, Larsen; Wilton, Henry. Stackings and the W-cycles Conjecture. Canadian mathematical bulletin, Tome 60 (2017) no. 3, pp. 604-612. doi: 10.4153/CMB-2016-078-3
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     title = {Stackings and the {W-cycles} {Conjecture}},
     journal = {Canadian mathematical bulletin},
     pages = {604--612},
     year = {2017},
     volume = {60},
     number = {3},
     doi = {10.4153/CMB-2016-078-3},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2016-078-3/}
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