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*:
*G* → *M*(*X*), where *M*(*X*)
is the group of all invertible maps *X* → *X*.

$ __Right action__: A map
*X* × *G* → *X*, (*x*, *g*)
\(\mapsto\) *x* · *g*, that satisfies,
∀*x* ∈ *X* 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* × *X* → *X*, the equivalence
classes *X*/~, where *x* ~ *x*' iff ∃ *g*
∈ *G* such that *gx* = *x*'.

* __Stabilizer of a point__:
Given some point *x* ∈ *X*, the subgroup of *G* of
all *g* such that *x**g* = *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*, *g* →
*σ*_{g}, 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 *σ*_{g}(*x*)
= *x* for all *x* ∈ *X* implies *g* = *e*.

* __Idea__: Only the identity leaves
all of *X* invariant, but some *g*s can have fixed points;
This condition is weaker than that defining free action.

**Free Action of a Group on a Manifold**

$ __Def__: The action
*σ*_{g}: *X*
→ *X* of a group *G* on a manifold *X* is free
iff only *e* has fixed points, i.e.,

*g* ∈ *G* & *g*
≠ *e* implies
*σ*_{g}(*x*)
≠ *x* , for all *x* ∈ *X* .

* __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
*x* ∈ *X* there is a neighborhood *U* of *x*
such that ∀*g* ∈ *G*, *g* ≠ *e*,
*U* ∩ *gU* = Ø (*X*/*G* ∈ Man);
(2, optional) if for some *p*, *q* ∈ *X* there
is no *g* ∈ *G* such that *gp* = *q*, then
there are neighborhoods *U* of *p* and *V* of *q*
such that ∀*g* ∈ *G*, *gU* ∩ *V*
= 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* × *X* → *X* is said to be transitive if any two *x*,
*y* ∈ *X* can be connected by a *g* ∈ *G*, i.e.,
for all *x*, *y* ∈ *X*, ∃ *g* ∈ *G* such
that *σ*_{g}(*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 JDG-m.DG/04 [Lichnerowicz-Obata conjecture].

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send feedback and suggestions to bombelli at olemiss.edu – modified 11 aug 2018