Representations of Galois Groups on the Homology of Surfaces

Abstract

Let $p:\Sigma'\to\Sigma$ be a finite Galois cover, possibly branched, with Galois group $G$. We are interested in the structure of the cohomology of $\Sigma'$ 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 $H_1(\Sigma',\bC)$. In the unbranched case, we algebro-geometrically realize the representation of $G$ on holomorphic 1-forms on $\Sigma'$ 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.