Linearization of Regular Proper Groupoids

TL;DR

This paper proves that proper Lie groupoids near finite type orbits can be linearized to action groupoids on normal bundles, using deformation and slice theorems.

Contribution

It introduces a new linearization theorem for proper Lie groupoids around finite type orbits, extending previous results with a novel proof technique.

Findings

Proper Lie groupoids are linearizable near finite type orbits.

The proof employs a cohomology vanishing theorem and a new slice theorem.

The results generalize classical linearization theorems for Lie group actions.

Abstract

Let G be a Lie groupoid over M such that the target-source map from G to M x M is proper. We show that, if O is an orbit of finite type (i.e. which admits a proper function with finitely many critical points), then the restriction G|U of G to some neighborhood U of O in M is isomorphic to a similar restriction of the action groupoid for the linear action of the transitive groupoid G|O on the normal bundle NO. 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.