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Given an action of a group on some space, and given a point or (or more generally some subspace), then the stabilizer group of that point (that subspace) is the subgroup whose action leaves the point (the subspace) fixed, invariant.
The importance of stabilizer subgroups for the general development of geometry was famously highlighted in (Klein 1872) in the context of what has come to be known the Erlangen program. For more on this aspect see at Klein geometry and Cartan geometry.
Given an action of a group on a set , for every element , the stabilizer subgroup of (also called the isotropy group of ) is the set of all elements in that leave fixed:
If all stabilizer groups are trivial, then the action is called a free action.
We reformulate the tradtitional definition above from the nPOV, in terms of homotopy theory.
A group action is equivalently encoded in its action groupoid fiber sequence in Grpd
where the is the action groupoid itself, is the delooping groupoid of and is regarded as a 0-truncated groupoid.
This fiber sequence may be thought of as being the -associated bundle to the -universal principal bundle. (Here discussed for a discrete group but this discussion goes through verbatim for a cohesive group).
any global element of , we have an induced element of the action groupoid and may hence form the first homotopy group . This is the stabilizer group. Equivalently this is the loop space object of at , given by the homotopy pullback
This characterization immediately generalizes to stabilizer ∞-groups of ∞-group actions. This we discuss below
For -group actions
Let be an (∞,1)-topos and be an ∞-group object in . Write for its delooping object.
By the discussion at ∞-action we have the following.
For any object, an ∞-action of on is equivalently an object and a homotopy fiber sequence of the form
Here is the homotopy quotient of the ∞-action
Given an ∞-action of on as in prop. 1, and given a global element of
then the stabilizer -group of the -action at is the loop space object
Equivalently, def. 1, gives the loop space object of the 1-image of the morphism
As such the delooping of the stabilizer -group sits in a 1-epimorphism/1-monomorphism factorization which combines with the homotopy fiber sequence of prop. 1 to a diagram of the form
In particular there is hence a canonical homomorphism of -groups
However, in contrast to the classical situation, this morphism is not in general a monomorphism anymore, hence the stabilizer is not a sub-group of in general.
For a group acting on itself
For any ∞-group in an (∞,1)-topos , its (right) action on itself is given by the looping/delooping fiber sequence
Clearly, for every point we have is trivial. Hence the action is free.
Stabilizers of shapes – Klein geometry
Let be an ∞-action of on .
Let any other object, and regard it as equipped with the trivial -action . There is then an induced ∞-action on the internal hom , the conjugation action, given by internal hom in the slice (∞,1)-topos over :
Now given any , then the stabilizer group is the stabilizer of “in” under this -action.
The morphism of -groups
hence characterizes the (higher) Klein geometry induced by the -action and by the shape . (See at Klein geometry – History.)
For completeness we notice that:
Here is equivalently the (∞,1)-pullback in
where the bottom morphism is the internal hom adjunct of the projection .
We check the Hom adjunction property, that for any we have
with replaced by the above pullback.
Notice that by the -action on being trivial, we have .
Then use the characterization of Hom-spaces in a slice to find as the homotopy pullback on the left of
Now using the Hom-adjunction in itself, the fact that preserves homotopy pullbacks and the pasting law this is equivalent to
Here the bottom map is indeed the name of and so again by the pullback characterization of Hom-spaces in a slice this pasting diagram does exhibit as indicated.
For the canonical action on a coset space
Conversely, for any homomorphism of ∞-groups given, then the canonical --action for which is the stabilizer -group of a point is the canonical action on (the “coset”) .
This follows from def. 1 by observing that the homotopy fiber sequence of prop. 1 for the -action on is just
so that for any point we have
Stabilizers of coshapes
Dually to “stabilizers of shapes”, as above one may consider stabilizers of “co-shapes”.
I.e. given a -action on , and given a map , then one may ask for the stabilizer of in the canonical -action on .
For instance if here is , and is regarded as a prequantum n-bundle ,and is replaced by its differential concretification, then these stabilizers are the quantomorphism n-groups.
to be expanded on
An early occurence of the concept of stabilizer subgroups is in p. 4 of
Felix Klein, Vergleichende Betrachtungen über neuere geometrische Forschungen (1872)
translation by M. W. Haskell, A comparative review of recent researches in geometry , trans. M. W. Haskell, Bull. New York Math. Soc. 2, (1892-1893), 215-249. (retyped pdf, retyped pdf, scan of original)
(the “Erlangen program”).