Free action

Let $G$ act on a manifold $M$.

The action is free if for every $x\in M$, the stabilizer is trivial:

Equivalently, the action is free if

For a smooth action of a Lie group, freeness implies each orbit is diffeomorphic to $G$ (via any choice of basepoint in the orbit), though generally not canonically.

Examples

1. Left translation. The action of a Lie group $G$ on itself by left multiplication is free.
2. Deck transformations. The action of $\mathbb{Z}$ on $\mathbb{R}$ by $n\cdot x := x+n$ is free.
3. Non-example (fixed point). The rotation action $SO(2)\curvearrowright \mathbb{R}^2$ is not free because every element fixes the origin.