group theory

# Contents

## 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:

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

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

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

where the $X \sslash G$ 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 $x\colon * \to X \to X \sslash G$ of the action groupoid and may hence form the first homotopy group $\pi_1(X \sslash G, x)$. This is the stabilizer group. Equivalently this is the loop space object of $X \sslash G$ at $x$, given by the homotopy pullback

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

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

### For $\infty$-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.

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

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

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

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

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

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

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

$i_x\colon Stab_\rho(x) \to G$

given by the looping of $\rho$

$i_x \coloneqq \Omega_x(\rho) \,.$

## Examples

### For an $\infty$-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

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

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

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

Then for $f\colon Y \to X$ a “shape” $Y$ in $X$, the stabilizer ∞-group of $Y$ under $\rho$ is $Stab_{\rho_Y}(f)$.

The morphism of $\infty$-groups

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

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

Revised on November 1, 2013 06:38:17 by Urs Schreiber (145.116.130.141)