Representations of Galois Groups on the Homology of Surfaces

Thomas Koberda; Aaron Michael Silberstein

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arXiv · 0905.3002

Representations of Galois Groups on the Homology of Surfaces

Thomas Koberda·Aaron Michael Silberstein

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Abstract

Let $p:Σ'\toΣ$ be a finite Galois cover, possibly branched, with Galois group $G$. We are interested in the structure of the cohomology of $Σ'$ as a module over $G$. We treat the cases of branched and unbranched covers separately. In the case of branched covers, we give a complete classification of possible module structures of $H_1(Σ',\bC)$. In the unbranched case, we algebro-geometrically realize the representation of $G$ on holomorphic 1-forms on $Σ'$ as functions on unions of $G$-torsors. We also explicitly realize these representations for certain branched covers of hyperelliptic curves. We give some applications to the study of pseudo-Anosov homeomorphisms of surfaces and representation theory of the mapping class group.

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BibTeX RIS

Thomas Koberda, Aaron Michael Silberstein. 2009-10-11. Representations of Galois Groups on the Homology of Surfaces. https://arxiv.org/abs/0905.3002

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