Contents

# Contents

## Idea

The continuous group action of a topological group $G$ is called “proper” if it behaves like the action of a compact topological group without $G$ necessarily being compact.

In particular, the isotropy groups of a proper action are compact.

## Definition

Consider

• a topological group $G$,

• a locally compact Hausdorff space $X$,

• a group action of $G$ on $X$ (making $X$ a topological G-space):

$\array{ G \times X &\overset{\rho}{\longrightarrow}& X \\ (x,g) &\mapsto& g \cdot x \mathrlap{\,.} }$

The action $\rho \colon (g,x) \mapsto g \cdot x$ is called proper (Palais 61, Def. 1.2.2) if one of the following equivalent conditions hold (their equivalence relies on $X$ being locally compact and Hausdorff, Palais 61, Thm. 1.2.9, Karppinen 16, Rem. 5.2.4):

1. (Bourbaki properness) The shear map is a proper continuous function:

$\array{ G \times X &\overset{proper}{\longrightarrow}& X \times X \\ (g,x) &\mapsto& (g\cdot x, x) }$

(Bourbaki 60, Ch. III, Sec. 4.4, review in Lee 00, p. 266)

2. (Borel properness) For every compact subspace $K \subset X$ the subset

$(K \vert K) \;\coloneqq\; \big\{ g \in G \,\vert\, g \cdot K \,\cap\, K \neq \varnothing \big\} \;\subset\; G$

is compact.

(Palais 61, Thm 1.2.9 (5), attributed there to Borel)

3. (Palais properness) Every point $x \in X$ has a neighbourhood $U_x$ such that every point $y \in U_x$ has a neighbourhood $V_y$ such that

$(U_x \vert U_y) \;\coloneqq\; \big\{ g \in G \,\vert\, g \cdot V_x \,\cap\, U_y \neq \varnothing \big\} \;\subset\; G$

has compact closure.

## Examples

### Lie group actions on smooth manifolds

How is the following not a trivial consequence of the fact that compact groups have proper action? Needs clarification.

###### Proposition

Let $X$ be a smooth manifold and let $G$ be a compact Lie group. Then every smooth action of $G$ on $X$ is proper.

(e.g. Lee 12, Cor. 7.2)

For more see at equivariant differential topology.

The original definition are due to:

Textbook accounts: