stabilizer group




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

Stab G(x)={gGgx=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 ρ:G×XX\rho\colon G \times X \to X is equivalently encoded in its action groupoid fiber sequence in Grpd

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

where the XGX \sslash G is the action groupoid itself, BG\mathbf{B}G is the delooping groupoid of GG and XX is regarded as a 0-truncated groupoid.

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


x:*X x\colon * \to X

any global element of XX, we have an induced element x:*XXGx\colon * \to X \to X \sslash G of the action groupoid and may hence form the first homotopy group π 1(XG,x)\pi_1(X \sslash G, x). This is the stabilizer group. Equivalently this is the loop space object of XGX \sslash G at xx, given by the homotopy pullback

Stab G(x) * x * x XG. \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 H\mathbf{H} be an (∞,1)-topos and GGrp(G)G \in \infty Grp(G) be an ∞-group object in H\mathbf{H}. Write BGH\mathbf{B}G \in \mathbf{H} for its delooping object.

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

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

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

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

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

Stab ρ(x)Ω x(XG). Stab_\rho(x) \coloneqq \Omega_x (X\sslash G) \,.

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

i x:Stab ρ(x)G i_x\colon Stab_\rho(x) \to G

given by the looping of ρ\rho

i xΩ x(ρ). i_x \coloneqq \Omega_x(\rho) \,.


For an \infty-group acting on itself

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

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

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

Stabilizers of shapes

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

[Y,X]G [Y,XG] ρ Y [Y,ρ] BG [Y,BG], \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×BGBGY \times \mathbf{B}G \to \mathbf{B}G.

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

The morphism of \infty-groups

i f:Stab ρ Y(f)G i_f\colon Stab_{\rho_Y}(f) \to G

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

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