Curves with prescribed symmetry and associated representations of mapping class groups

Abstract Let C be a complex smooth projective algebraic curve endowed with an action of finite group G such that the quotient has genus at least 3. We prove if -curve is very general for these properties, then natural map from algebra $${{\mathbb {Q}}}G$$ Q G to {Q}}}$$ -endomorphisms its Jacobian isomorphism. use this obtain (topological) properties regarding certain virtual linear representations mapping class group. For example, we show connected component Zariski closure representation often acts -irreducibly in -isogeny space $$H^1(C; {{\mathbb {Q}}})$$ H 1 ( C ; ) and image -almost simple