Strong symmetric genus of a finite group

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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