# Strong symmetric genus of a finite group

From Groupprops

## Definition

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

- It is the smallest genus of a compact connected oriented surface on which acts faithfully via orientation-preserving diffeomorphisms.
- It is the smallest genus of a compact connected Riemann surface on which acts faithfully via Riemann surface isomorphisms, i.e., conformal mappings.
- it is the smallest genus of a compact connected two-dimensional Riemannian manifold on which 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 is more than one, then it is at least . 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.