nLab stabilizer group

Idea

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.

Sometimes (such as in the context of Wigner classification) stabilizer groups are called little groups.

Definition

Traditional

Given an action $G\times X\to X$ of a group $G$ on a set $X$, for every element $x \in X$, the stabilizer subgroup of $x$ (also called the isotropy group of $x$) is the set of all elements in $G$ that leave $x$ fixed:

$$Stab_G(x) = \{g \in G \mid g\circ x = x\} \,.$$

If all stabilizer groups are trivial, then the action is called a free action.

Homotopy-theoretic formulation

We reformulate the traditional definition above from the nPOV, in terms of homotopy theory.

A group action $\rho\colon G \times X \to X$ is equivalently encoded in its action groupoid fiber sequence in Grpd

$$X \to X//G \to \mathbf{B}G \,,$$

where the $X//G$ is the action groupoid itself, $\mathbf{B}G$ is the delooping groupoid of $G$ and $X$ is regarded as a 0-truncated groupoid.

This fiber sequence may be thought of as being the $\rho$-associated bundle to the $G$-universal principal bundle. (Here discussed for $G$ a discrete group but this discussion goes through verbatim for $G$ a cohesive group.)

For

$$x\colon * \to X$$

any global element of $X$, we have an induced element $x\colon * \to X \to X//G$ of the action groupoid and may hence form the first homotopy group $\pi_1(X//G, x)$. This is the stabilizer group. Equivalently this is the loop space object of $X//G$ at $x$, given by the homotopy pullback

$$\array{ Stab_G(x) &\to& * \\ \downarrow && \downarrow^{\mathrlap{x}} \\ * &\stackrel{x}{\to}& X//G } \,.$$

This characterization immediately generalizes to stabilizer ∞-groups of ∞-group actions. This we discuss below

For $\infty$-group actions

Let $\mathbf{H}$ be an (∞,1)-topos and $G \in \infty Grp(G)$ be an ∞-group object in $\mathbf{H}$. Write $\mathbf{B}G \in \mathbf{H}$ for its delooping object.

By the discussion at ∞-action we have the following.

Proposition

For $X \in \mathbf{H}$ any object, an ∞-action of $G$ on $X$ is equivalently an object $X/G$ and a homotopy fiber sequence of the form

$$\array{ X &\longrightarrow& X//G \\ && \downarrow \\ && \mathbf{B}G } \,.$$

Here $X/G$ is the homotopy quotient of the ∞-action

Remark

The action as a morphism $X \times G \to X$ is recovered from prop. by the (∞,1)-pullback

$$\array{ X \times G &\to& X \\ \downarrow && \downarrow \\ X &\to& X//G } \,.$$

Definition

Given an ∞-action $\rho$ of $G$ on $X$ as in prop. , and given a global element of $X$

$$x \colon \ast \to X$$

then the stabilizer $\infty$-group $Stab_\rho(x)$ of the $G$-action at $x$ is the loop space object

$$Stab_\rho(x) \coloneqq \Omega_x (X//G) \,.$$

Remark

Equivalently, def. , gives the loop space object of the 1-image $\mathbf{B}Stab_\rho(x)$ of the morphism

$$\ast \stackrel{x}{\to} X \to X//G \,.$$

As such the delooping of the stabilizer $\infty$-group sits in a 1-epimorphism/1-monomorphism factorization $\ast \to \mathbf{B}Stab_\rho(x) \hookrightarrow X//G$ which combines with the homotopy fiber sequence of prop. to a diagram of the form

$$\array{ \ast &\stackrel{x}{\longrightarrow}& X &\stackrel{}{\longrightarrow}& X//G \\ \downarrow^{\mathrlap{epi}} && & \nearrow_{\mathrlap{mono}} & \downarrow \\ \mathbf{B} Stab_\rho(x) &=& \mathbf{B} Stab_\rho(x) &\longrightarrow& \mathbf{B}G } \,.$$

In particular there is hence a canonical homomorphism of $\infty$-groups

$$Stab_\rho(x) \longrightarrow G \,.$$

However, in contrast to the classical situation, this morphism is not in general a monomorphism anymore, hence the stabilizer $Stab_\rho(x)$ is not a sub-group of $G$ in general.

Examples

For a group acting on itself

For $G$ any ∞-group in an (∞,1)-topos $\mathbf{H}$, its (right) action on itself is given by the looping/delooping fiber sequence

$$G \to * \stackrel{\rho}{\to} \mathbf{B}G \,.$$

Clearly, for every point $g \in G$ we have $Stab_{\rho}(g) \simeq * \times_* * \simeq *$ is trivial. Hence the action is free.

Stabilizers of shapes – Klein geometry

Let $X \to X//G \stackrel{\rho}{\to} \mathbf{B}G$ be an ∞-action of $G$ on $X$.

Let $Y \in \mathbf{H}$ any other object, and regard it as equipped with the trivial $G$-action $Y \to Y \times \mathbf{B}G \to \mathbf{B}G$. There is then an induced ∞-action $\rho_Y$ on the internal hom $[Y,X]$, the conjugation action, given by internal hom in the slice (∞,1)-topos over $\mathbf{B}G$:

$$[Y,X] \to \underset{\mathbf{B}G}{\sum} [Y,X]_{/\mathbf{B}G} \to \mathbf{B}G \,.$$

Now given any $f \colon Y \to X$, then the stabilizer group $Stab_{\rho_Y}(f)$ is the stabilizer of $Y$ “in” $X$ under this $G$-action.

The morphism of $\infty$-groups

$$i_f\colon Stab_{\rho_Y}(f) \to G$$

hence characterizes the (higher) Klein geometry induced by the $G$-action and by the shape $f\colon Y \to X$. (See at Klein geometry – History.)

For completeness we notice that:

Proposition

Here $(\underset{\mathbf{B}G}{\sum} [Y,X]_{/\mathbf{B}G} \to \mathbf{B}G )$ is equivalently the (∞,1)-pullback $\rho_Y$ in

$$\array{ \underset{\mathbf{B}G}{\sum} [Y,X]_{/\mathbf{B}G} &\to& [Y, X//G] \\ \downarrow^{\mathrlap{\rho_Y}} && \downarrow^{\mathrlap{[Y, \rho]}} \\ \mathbf{B}G &\to& [Y, \mathbf{B}G] } \,,$$

where the bottom morphism is the internal hom adjunct of the projection $Y \times \mathbf{B}G \to \mathbf{B}G$.

Proof

We check the Hom adjunction property, that for any $(A//G \stackrel{\alpha}{\to} \mathbf{B}G) \in \mathbf{B}G$ we have

$$\mathbf{H}_{/\mathbf{B}G}(A,[Y,X]_{\mathbf{B}G}) \simeq \mathbf{H}_{/\mathbf{B}G}(A\times_{\mathbf{B}G} Y, X)$$

with $[Y,X]_{/\mathbf{B}G}$ replaced by the above pullback.

Notice that by the $G$-action on $Y$ being trivial, we have $A \times_{\mathbf{B}G} Y \simeq (A//G \times Y \stackrel{p_1}{\to} A//G \stackrel{\alpha}{\to} \mathbf{B}G) \in \mathbf{H}_{/\mathbf{B}G}$.

Then use the characterization of Hom-spaces in a slice to find $\mathbf{H}_{/\mathbf{B}G}(A,[Y,X]_{\mathbf{B}G})$ as the homotopy pullback on the left of

$$\array{ \mathbf{H}_{/\mathbf{B}G}(A,[Y,X]_{\mathbf{B}G}) &\longrightarrow& \mathbf{H}(A//G, [Y,X]//G) &\to& \mathbf{H}(A//G,[Y, X//G]) \\ \downarrow && \downarrow^{\mathrlap{\mathbf{H}(A//G,\rho_Y)}} && \downarrow^{\mathrlap{\mathbf{H}(A//G,[Y, \rho])}} \\ \ast &\stackrel{\vdash \alpha}{\longrightarrow}& \mathbf{H}(A//G,\mathbf{B}G) &\to& \mathbf{H}(A//G,[Y, \mathbf{B}G]) } \,,$$

Now using the Hom-adjunction in $\mathbf{H}$ itself, the fact that $\mathbf{H}(A//G,-)$ preserves homotopy pullbacks and the pasting law this is equivalent to

$$\array{ \mathbf{H}_{/\mathbf{B}G}(A\times_{\mathbf{B}G} Y, X) &\longrightarrow& &\longrightarrow& \mathbf{H}(A//G \times Y, X //G) \\ \downarrow && \downarrow && \downarrow \\ \ast &\stackrel{\vdash \alpha}{\longrightarrow}& \mathbf{H}(A//G,\mathbf{B}G) &\to& \mathbf{H}(A//G \times Y, \mathbf{B}G) } \,,$$

Here the bottom map is indeed the name of $\alpha \circ p_1$ and so again by the pullback characterization of Hom-spaces in a slice this pasting diagram does exhibit $\mathbf{H}_{/\mathbf{B}G}(A\times_{\mathbf{B}G} Y, X)$ as indicated.

For the canonical action on a coset space

Conversely, for any homomorphism $H \to G$ of ∞-groups given, then the canonical $G$-$\infty$-action for which $H$ is the stabilizer $\infty$-group of a point is the canonical action on (the “coset”) $G/H$.

This follows from def. by observing that the homotopy fiber sequence of prop. for the $G$-action on $G/H$ is just

$$\array{ G/H &\stackrel{}{\longrightarrow}& \mathbf{B}H \\ && \downarrow \\ && \mathbf{B}G }$$

so that for any point $x \colon \ast \to G/H$ we have

$$Stab(x) \simeq \Omega_{x}(\mathbf{B}H) \simeq \Omega_\ast \mathbf{B}H \simeq H \,.$$

Stabilizers of coshapes

Dually to “stabilizers of shapes”, as above one may consider stabilizers of “co-shapes”.

I.e. given a $G$-action on $X$, and given a map $f \colon X \to A$, then one may ask for the stabilizer of $f$ in the canonical $G$-action on $[X,A]$.

For instance if $A$ here is $\mathbf{B}^{n}U(1)_{conn}$, and $f \colon X \to \mathbf{B}^n U(1)_{conn}$ is regarded as a prequantum n-bundle ,and $[X,A]$ is replaced by its differential concretification, then these stabilizers are the quantomorphism n-groups.

to be expanded on

Related concepts

• normalizer subgroup

• centralizer subgroup

• stable point

• Klein geometry, Cartan geometry

References

An early occurence of the concept of stabilizer subgroups:

• Felix Klein, p. 4 of: 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”).

• Glen Bredon, Section I.2 of: Introduction to compact transformation groups, Academic Press 1972 (ISBN 9780080873596, pdf)