We prove that the number of distinct group actions on compact Riemann surfaces of a fixed genus is at least quadratic in . We do this through the introduction of a coarse signature space, the space of skeletal signatures of group actions on compact Riemann surfaces of genus . We discuss the basic properties of and present a full conjectural description.
James W Anderson. Aaron Wootton. "A lower bound for the number of group actions on a compact Riemann surface." Algebr. Geom. Topol. 12 (1) 19 - 35, 2012. https://doi.org/10.2140/agt.2012.12.19