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 $\sigma \ge 2$ is at least quadratic in $\sigma$. We do this through the introduction of a coarse signature space, the space ${\mathcal{K}}_{\sigma }$ of {\em skeletal signatures} of group actions on compact Riemann surfaces of genus $\sigma$. We discuss the basic properties of ${\mathcal{K}}_{\sigma }$ and present a full conjectural description.