Desingularization of Lie groupoids and pseudodifferential operators on singular spaces

TL;DR

This paper introduces a method to desingularize Lie groupoids along specific submanifolds, enabling better analysis on singular spaces by explicitly describing the structure and Lie algebroid of the resulting groupoid.

Contribution

It develops a canonical desingularization process for Lie groupoids along A(G)-tame submanifolds, with explicit structure and applications to pseudodifferential operators on singular spaces.

Findings

Constructed the desingularization groupoid [[G:L]] with explicit structure.

Identified the Lie algebroid of the desingularized groupoid.

Connected the construction to edge pseudodifferential calculus.

Abstract

We introduce and study a "desingularization" of a Lie groupoid $G$ along an "$A(G)$-tame" submanifold $L$ of the space of units $M$. An $A(G)$-tame submanifold $L \subset M$ is one that has, by definition, a tubular neighborhood on which $A(G)$ becomes a thick pull-back Lie algebroid. The construction of the desingularization $[[G:L]]$ of $G$ along $L$ is based on a canonical fibered pull-back groupoid structure result for $G$ in a neighborhood of the tame $A(G)$-submanifold $L \subset M$. This local structure result is obtained by integrating a certain groupoid morphism, using results of Moerdijk and Mrcun (Amer. J. Math. 2002). Locally, the desingularization $[[G:L]]$ is defined using a construction of Debord and Skandalis (Advances in Math., 2014). The space of units of the desingularization $[[G:L]]$ is $[M:L]$, the blow up of $M$ along $L$. The space of units and the desingularization groupoid $[[G:L]]$ are constructed using a gluing construction of Gualtieri and Li (IMRN 2014). We provide an explicit description of the structure of the desingularized groupoid and we identify its Lie algebroid, which is important in analysis applications. We also discuss a variant of our construction that is useful for analysis on asymptotically hyperbolic manifolds. We conclude with an example relating our constructions to the so called "edge pseudodifferential calculus." The paper is written such that it also provides an introduction to Lie groupoids designed for applications to analysis on singular spaces.