Canonical stratification of definable Lie groupoids

Abstract

Our aim is to precisely present a tame topology counterpart to canonical stratification of a Lie groupoid. We consider a definable Lie groupoid in semialgebraic, subanalytic, o-minimal over $\mathbb{R}$, or more generally, Shiota's $\mathfrak{X}$-category. We show that there exists a canonical Whitney stratification of the Lie groupoid into definable strata which are invariant under the groupoid action. This is a generalization and refinement of results on real algebraic group action which J. N. Mather and V. A. Vassiliev independently stated with sketchy proofs. A crucial change to their proofs is to use Shiota's isotopy lemma and approximation theorem in the context of tame topology.