nLab
stabilizer group

Definition

Explicitly

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:

$${\mathrm{Stab}}_G(x)=\{g\in G\mid g\circ x=x\}.$$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.

General abstract characterization

We discuss stabilizer subgroups from the nPOV.

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

$$X\to X⫽G\to BG,$$X \to X \sslash G \to \mathbf{B}G \,,

where the $X⫽G$ is the action groupoid itself, $BG$ 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

$$x:*\to X$$x\colon * \to X

any global element of $X$, we have an induced element $x:*\to X\to X⫽G$ 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

$$\begin{array}{ccc}{\mathrm{Stab}}_G(x)& \to & *\\ \downarrow & & {\downarrow }^x\\ *& \stackrel{x}{\to }& X⫽G\end{array}.$$\array{ Stab_G(x) &\to& * \\ \downarrow && \downarrow^{\mathrlap{x}} \\ * &\stackrel{x}{\to}& X\sslash G } \,.

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

For $\mathrm{\infty }$-group actions

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

Then for $X\in H$ any other object, an action of $G$ on $X$ is an object $X⫽G$ and a fiber sequence of the form

$$X\to X⫽G\stackrel{\rho }{\to }BG.$$X \to X \sslash G \stackrel{\rho}{\to} \mathbf{B}G \,.

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

$$\begin{array}{ccc}X\times G& \to & X\\ \downarrow & & \downarrow \\ X& \to & X⫽G\end{array}.$$\array{ X \times G &\to& X \\ \downarrow && \downarrow \\ X &\to& X \sslash G } \,.

Now for $x:*\to X$ any global element, the stabilizer $\mathrm{\infty }$-group of $\rho$ at $x$ is the loop space object

$${\mathrm{Stab}}_{\rho }(x)≔{\Omega }_x(X⫽G).$$Stab_\rho(x) \coloneqq \Omega_x (X\sslash G) \,.

This is equipped with a canonical morphism of ∞-group objects

$$i_x:{\mathrm{Stab}}_{\rho }(x)\to G$$i_x\colon Stab_\rho(x) \to G

given by the looping of $\rho$

$$i_x≔{\Omega }_x(\rho ).$$i_x \coloneqq \Omega_x(\rho) \,.

Examples

For an $\mathrm{\infty }$-group acting on itself

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

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

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

Stabilizers of shapes

For $X⫽G\stackrel{\rho }{\to }BG$ an action, and $Y\in H$ any other object, we get an induced action ${\rho }_Y$ on the internal hom $[Y,X]$ defined as the (∞,1)-pullback

$$\begin{array}{ccc}[Y,X]⫽G& \to & [Y,X⫽G]\\ {\downarrow }^{{\rho }_Y}& & {\downarrow }^{[Y,\rho ]}\\ BG& \to & [Y,BG]\end{array},$$\array{ [Y,X] \sslash G &\to& [Y, X \sslash 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 $Y\times BG\to BG$.

Then for $f:Y\to X$ a “shape” $Y$ in $X$, the stabilizer ∞-group of $Y$ under $\rho$ is ${\mathrm{Stab}}_{{\rho }_Y}(f)$.

The morphism of $\mathrm{\infty }$-groups

$$i_f:{\mathrm{Stab}}_{{\rho }_Y}(f)\to G$$i_f\colon Stab_{\rho_Y}(f) \to G

characterizes the higher Klein geometry induced by $f:Y\to X$.