Finite covers of graphs, their primitive homology, and representation theory

Benson Farb; Sebastian Hensel

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arXiv · 1610.08819

Finite covers of graphs, their primitive homology, and representation theory

Benson Farb·Sebastian Hensel

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Abstract

Consider a finite, regular cover $Y\to X$ of finite graphs, with associated deck group $G$. We relate the topology of the cover to the structure of $H_1(Y;\mathbb{C})$ as a $G$-representation. A central object in this study is the {\em primitive homology} group $H_1^{\mathrm{prim}}(Y;\mathbb{C})\subseteq H_1(Y;\mathbb{C})$, which is the span of homology classes represented by components of lifts of primitive elements of $\pi_1(X)$. This circle of ideas relates combinatorial group theory, surface topology, and representation theory.

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Benson Farb, Sebastian Hensel. 2016-10-27. Finite covers of graphs, their primitive homology, and representation theory. https://arxiv.org/abs/1610.08819

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