nLab effective group action

This entry is about the concept in group theory. For the concept in quantumfield theory see at effective action functional; for disambiguation see effective action.

Idea

A group action is effective if no group element other than the neutral element acts trivially on all elements of the space, hence if no other element acts the way the neutral element does.

Definition

A group action of a group (group object) $G$ on a set (object) $X$ is effective if $\underset{x \in X}{\forall} g x = x$ implies that $g = e$ is the neutral element.

Beware the similarity to and difference with free action: a free action is effective, but an effective action need not be free.

Properties

Proposition

(Newman’s theorem) Given an effective action of a finite group $G$ on a connected manifold $X$ by homeomorphisms, then the subset $X_{reg} \subset X$ of points with trivial stabilizer group is open and everywhere dense.

(Newman 1931, cf. Dress 1969 Thm. 1)

Related concepts

• transitive action, free action, semi-free action, regular action,

• proper action, properly discontinuous action

• effective orbifold

References

The original reference for Newman’s theorem:

• M. H. A. Newman, A theorem on periodic transformations of spaces, The Quarterly Journal of Mathematics os-2:1 (1931), 1-8. DOI.

Streamlined review:

• Andreas Dress: Newman’s theorems on transformation groups, Topology 8 2 (1969) 203-207 [doi:10.1016/0040-9383(69)90010-X]