Fredholm criteria for pseudodifferential operators and induced representations of groupoid algebras

TL;DR

This paper characterizes when pseudodifferential operators on certain groupoids are Fredholm, linking invertibility of principal symbols and boundary restrictions, with applications to various geometric settings.

Contribution

It introduces the concept of Fredholm groupoids, characterizes them via groupoid properties, and applies the theory to diverse geometric contexts including manifolds with ends and desingularizations.

Findings

Almost amenable, Hausdorff, second countable groupoids are Fredholm.

Many practical pseudodifferential operators are covered by this framework.

Desingularization preserves the class of Fredholm groupoids.

Abstract

We characterize the groupoids for which an operator is Fredholm if, and only if, its principal symbol and all its boundary restrictions are invertible. A groupoid with this property is called {\em Fredholm}. Using results on the Effros-Hahn conjecture, we show that an almost amenable, Hausdorff, second countable groupoid is Fredholm. Many groupoids, and hence many pseudodifferential operators appearing in practice, fit into this framework. In particular, one can use these results to characterize the Fredholm operators on manifolds with cylindrical and poly-cylindrical ends, on manifolds that are asymptotically Euclidean or asymptotically hyperbolic, on products of such manifolds, and on many others. Moreover, we show that the desingularization of Lie groupoids preserves the class of Fredholm groupoids.