Action
of a Group G on a Set X| Action
of a Group G on a Set X |
In General > s.a. [lie group]; Homogeneous
Space; representations.
$ Def: A homomorphism 
:
G → M(X), where M(X)
is the group of all invertible maps X → X.
$ Right action: A map X × G → X,
(x, g) 
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 × F → F,
the equivalence classes F/
,
where f f '
iff 
g G such
that gf = f '.
* 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 
:
G → Diff(X)
from the group to the diffeomorphisms of a manifold X, g → 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.
Effective Action of a Group on a Manifold
$ Def: G acts
effectively on X if 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 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 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 thatg 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 thatg 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 : 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 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 m.DG/04 [Lichnerowicz-Obata conjecture].
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2009