Strong symmetric genus of a finite group

Definition
The strong symmetric genus of a finite group $$G$$, sometimes denoted $$\sigma^{\circ}(G)$$, is defined in the following equivalent ways:


 * 1) It is the smallest genus $$\sigma^\circ$$ of a compact connected oriented surface on which $$G$$ acts faithfully via orientation-preserving diffeomorphisms.
 * 2) It is the smallest genus $$\sigma^\circ$$ of a compact connected Riemann surface on which $$G$$ acts faithfully via Riemann surface isomorphisms, i.e., conformal mappings.
 * 3) it is the smallest genus $$\sigma^\circ$$ of a compact connected two-dimensional Riemannian manifold on which $$G$$ acts faithfully via orientation-preserving isometries of the Riemannian metric.

The equivalence of these essentially follows from the fact that any action of type (1) gives an action of type (3) by choosing a Riemannian metric by averaging. Type (2) is in between.

Facts

 * If the strong symmetric genus of a group $$G$$ is more than one, then it is at least $$1 + (|G|/84)$$. Groups for which equality holds are called Hurwitz groups.

Related notions

 * Symmetric genus of a finite group is the corresponding notion where we do not require the diffeomorphisms to be orientation-preserving.