On invariant linearization of Lie groupoids

TL;DR

This paper advances the understanding of invariant linearization of proper Lie groupoids by providing a counter-example, a new sufficient criterion, and extending previous results, thereby clarifying the conditions under which linearization is possible.

Contribution

It introduces a counter-example to a prior conjecture and establishes a new sufficient criterion for invariant linearization using compatible complete metrics.

Findings

Counter-example to previous conjecture

Sufficient criterion for linearization with compatible metrics

Extension of previous results in the literature

Abstract

The Linearization Theorem for proper Lie groupoids organizes and generalizes several results for classic geometries. Despite the various approaches and recent works on the subject, the problem of understanding invariant linearization remains somehow open. We address it here, by first giving a counter-example to a previous conjecture, and then proving a sufficient criterion that uses compatible complete metrics and covers the case of proper group actions. We also show a partial converse that fixes and extends previous results in the literature.