Strong symmetric genus of a finite group: Difference between revisions
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==Definition== | ==Definition== | ||
The '''strong symmetric genus''' of a [[finite group]] <math>G</math> is defined in the following equivalent ways: | The '''strong symmetric genus''' of a [[finite group]] <math>G</math>, sometimes denoted <math>\sigma^{\circ}(G)</math>, is defined in the following equivalent ways: | ||
# It is the smallest genus <math>\sigma</math> of a compact connected oriented surface on which <math>G</math> acts faithfully via orientation-preserving diffeomorphisms. | # It is the smallest genus <math>\sigma^\circ</math> of a compact connected oriented surface on which <math>G</math> acts faithfully via orientation-preserving diffeomorphisms. | ||
# It is the smallest genus <math>\sigma</math> of a compact connected Riemann surface on which <math>G</math> acts faithfully via Riemann surface isomorphisms, i.e., conformal mappings. | # It is the smallest genus <math>\sigma^\circ</math> of a compact connected Riemann surface on which <math>G</math> acts faithfully via Riemann surface isomorphisms, i.e., conformal mappings. | ||
# it is the smallest genus <math>\sigma</math> of a compact connected two-dimensional Riemannian manifold on which <math>G</math> acts faithfully via orientation-preserving isometries of the Riemannian metric. | # it is the smallest genus <math>\sigma^\circ</math> of a compact connected two-dimensional Riemannian manifold on which <math>G</math> 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. | 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 <math>G</math> is more than one, then it is at least <math>1 + (|G|/84)</math>. Groups for which equality holds are called [[Hurwitz group]]s. | |||
==Related notions== | ==Related notions== | ||
* [[Symmetric genus of a finite group]] is the corresponding notion where we do not require the diffeomorphisms to be orientation-preserving. | * [[Symmetric genus of a finite group]] is the corresponding notion where we do not require the diffeomorphisms to be orientation-preserving. | ||
Latest revision as of 04:05, 25 December 2012
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.