Alan Weinstein
Linearization of Regular Proper Groupoids
Preprint series: ESI preprints

MSC:

58H05 Pseudogroups and differentiable groupoids, See also {22A22,

57R99 None of the above but in this section

Abstract: 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
neighborhood ${\mathcal U}$ 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 $N\calo$. 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.

Keywords: Lie groupoid, proper action, submersion