Abstract
Let S be a closed orientable surface of genus at least 2 and let \(\widetilde S\) to S be a connected finite abelian covering with covering group $G$. The lifts of liftable mapping classes of S determine a central extension (by G) of a subgroup of finite index of the mapping class group of S. This extension acts on H1(\(\widetilde S\)). With a few exceptions for genus 2, we determine the Zariski closure of the image of this representation, and prove that the image is an arithmetic group.
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Looijenga, E. Prym Representations of Mapping Class Groups. Geometriae Dedicata 64, 69–83 (1997). https://doi.org/10.1023/A:1004909416648
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DOI: https://doi.org/10.1023/A:1004909416648