Group Actions  

In General > s.a. lie group / Homogeneous Space; representations.
$ Def: An action of a group G on a set X is a homomorphism θ: GM(X), where M(X) is the group of all invertible maps XX.
$ Right action: A map X × GX, (x, g) \(\mapsto\) x · g, that satisfies, ∀xX and ∀g, g' ∈ G, (i) x · e = x; (ii) (x · g) · g' = x · (gg').
* Remark: Left action is analogous; The distinction between left and right is meaningful only when G is non-Abelian.
* Orbits: Given a group action G × XX, the equivalence classes X/~, where x ~ x' iff ∃ gG such that gx = x'.
* Stabilizer of a point: Given some point xX, the subgroup of G of all g such that xg = x (in right-action notation).
* Examples: Of a Lie group on itself by left-multiplication (free), or conjugation (not free); Of the Poincaré group on Minkowski spacetime (not free); Of a Lie group on a manifold (a realization, usually required to be a smooth diffeomorphism); On a vector space (a representation).
* Realization of a group: A mapping σ: G → Diff(X) from the group to the diffeomorphisms of a manifold X, gσg, which is a homomorphism, i.e., a smooth group action on a manifold; Special cases: The realization is faithful if the mapping is injective.
@ General references: in Reid 70 [on Klein's program for geometry]; Rudolph & Weiss AM(00)m.DS [amenable groups, entropy and mixing].
@ Non-linear realizations: Isham et al AP(71) [of spacetime symmetries]; > s.a. types of yang-mills theories.
> Online resources: see MathWorld page; Wikipedia page.

Effective Action of a Group on a Manifold
$ Def: G acts effectively on X if σg(x) = x for all xX implies g = e.
* Idea: Only the identity leaves all of X invariant, but some gs can have fixed points; This condition is weaker than that defining free action.

Free Action of a Group on a Manifold
$ Def: The action σg: XX of a group G on a manifold X is free iff only e has fixed points, i.e.,

gG & ge   implies   σg(x) ≠ x ,  for all xX .

* Example: The left action of a group on itself.
* Relationships: The condition is stronger than the one for an effective action.

Proper Discontinuous Action of a Group
$ Def: A group G acts properly discontinuously on a topological space X if (1) For all xX there is a neighborhood U of x such that ∀gG, ge, UgU = Ø (X/G ∈ Man); (2, optional) if for some p, qX there is no gG such that gp = q, then there are neighborhoods U of p and V of q such that ∀gG, gUV = Ø (X/G is Hausdorff).
* Example: Any finite group acting without fixed points on a Hausdorff space acts properly discontinuously.

Transitive Action of a Group on a Manifold
$ Def: A group action σ: G × XX is said to be transitive if any two x, yX can be connected by a gG, i.e., for all x, yX, ∃ gG such that σg(x) = y.
* Special case: If this g is unique, the action is called simply transitive.

Group Actions on Manifolds with Other Structure
@ Preserving geodesics: Matveev m.DG/04 [Lichnerowicz-Obata conjecture].

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