Let $G$ be a Lie groupoid over $M$ such that the target-source map from $G$ to $M\times M$ is proper. We show that, if $\mathcal{O}$ is an orbit of finite type (i.e. which admits a proper function with finitely many critical points), then the restriction $G|_\mathcal{U}$ of $G$ to some neighbourhood $\mathcal{U}$ of $\mathcal{O}$ in $M$ is isomorphic to a similar restriction of the action groupoid for the linear action of the transitive groupoid $G|_\mathcal{O}$ on the normal bundle $N\mathcal{O}$. The proof uses a deformation argument based on a cohomology vanishing theorem, along with a slice theorem which is derived from a new result on submersions with a fibre of finite type.

AMS 2000 Mathematics subject classification: Primary 58H05. Secondary 57R99