Symmetric genus of a finite group

Definition

The symmetric genus of a finite group $G$, denoted $\sigma(G)$, is defined in the following equivalent ways:

1. It is the smallest genus $\sigma$ of a compact connected oriented surface on which $G$ acts faithfully via diffeomorphisms, which may be orientation-preserving or orientation-reserving.
2. It is the smallest genus $\sigma$ of a compact connected Riemann surface on which $G$ acts faithfully via Riemann surface isomorphisms or anti-isomorphisms, i.e., by mappings that are either conformal or anti-conformal (i.e., they reverse the roles of $i,-i$).
3. it is the smallest genus $\sigma$ of a compact connected two-dimensional Riemannian manifold on which $G$ acts faithfully via 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.

Related notions

• Strong symmetric genus of a finite group is the version where we require the action to be orientation-preserving (in all the equivalent definitions).