Finite simple groups acting with fixity 4 and their occurrence as groups of automorphisms of Riemann surfaces (extended version)

In previous work, all finite simple groups that act with fixity 4 have been classified. In this article we investigate which ones of these groups act faithfully on a compact Riemann surface of genus at least 2 with fixity four in total and in such a way that fixity 4 is exhibited on at least one orbit. This is an extended version of the submitted article, including our GAP code.

💡 Research Summary

The paper investigates which finite simple groups that have already been classified as acting with fixity 4 can actually occur as groups of automorphisms of compact Riemann surfaces of genus ≥ 2, under the additional requirement that the total fixity on the surface does not exceed 4 and that at least one orbit exhibits fixity 4. The authors build on their earlier classification of simple groups with fixity 2, 3, 4 and now focus on the “realizable” cases in the geometric setting of Riemann surfaces.

The central notion is the fixity of a group action: for a faithful transitive action of a group G on a finite set Ω, the fixity k is the maximal number of points fixed by any non‑trivial element of G. The paper restricts attention to k = 4. In the context of Riemann surfaces, point stabilisers are always cyclic, and the action is described by a Hurwitz datum \