A lower bound for the number of group actions on a compact riemann surface

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Description/Abstract

We prove that the number of distinct group actions on compact Riemann surfaces of a fixed genus g (at least 2) is at least quadratic in g. We do this through the introduction of a coarse signature space, the space K_g of skeletal signatures of group actions on compact Riemann surfaces of genus g. We discuss the basic properties of K_g and present a full conjectural description.