A note on the global structure of proper Lie groupoids in low codimensions

TL;DR

This paper studies the global structure of connected proper Lie groupoids with low codimension orbits, showing they admit effective representations and can be described as extensions of action groupoids by bundles of compact Lie groups.

Contribution

It demonstrates that such Lie groupoids have globally effective representations and can be characterized as extensions of action groupoids by compact Lie group bundles.

Findings

Connected proper Lie groupoids with codimension ≤ 2 orbits admit effective representations.

Such groupoids are Morita equivalent to extensions of action groupoids by compact Lie group bundles.

The results provide a structural understanding of low codimension proper Lie groupoids.

Abstract

We observe that any connected proper Lie groupoid whose orbits have codimension at most two admits a globally effective representation on a smooth vector bundle, i.e., one whose kernel consists only of ineffective arrows. As an application, we deduce that any such groupoid can up to Morita equivalence be presented as an extension of some action groupoid G n X with G compact by some bundle of compact Lie groups.