A Lower Bound for the Number of Group Actions on a Compact Riemann Surface

We prove that the number of distinct group actions on compact Riemann surfaces of a fixed genus $σ\geq 2$ is at least quadratic in $σ$. We do this through the introduction of a coarse signature space, the space $\mathcal{K}_σ$ of {\em skeletal signatures} of group actions on compact Riemann surfaces of genus $σ$. We discuss the basic properties of $\mathcal{K}_σ$ and present a full conjectural description.

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