# Symmetric genus of a finite group

## Definition

The **symmetric genus** of a finite group , denoted , is defined in the following equivalent ways:

- It is the smallest genus of a compact connected oriented surface on which acts faithfully via diffeomorphisms, which may be orientation-preserving or orientation-reserving.
- It is the smallest genus of a compact connected Riemann surface on which 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 ).
- it is the smallest genus of a compact connected two-dimensional Riemannian manifold on which 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).