Stackings and the W-cycle conjecture

Abstract

We prove Wise's $W$-cycles conjecture. Consider a compact graph ${\Gamma }^{\prime }$ immering into another graph $\Gamma$. For any immersed cycle $\Lambda :S^1\to \Gamma$, we consider the map ${\Lambda }^{\prime }$ from the circular components $\mathbb{S}$ of the pullback to ${\Gamma }^{\prime }$. Unless ${\Lambda }^{\prime }$ is reducible, the degree of the covering map $\mathbb{S}\to S^1$ is bounded above by minus the Euler characteristic of ${\Gamma }^{\prime }$. As a consequence, we obtain a homological version of coherence for one-relator groups.

9 pages, 3 figures

Cited by in corpus (3)

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• Counting cycles in labeled graphs: The nonpositive immersion property for one-relator groups