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.