# Fredholm conditions on non-compact manifolds: theory and examples

## TL;DR

This paper establishes explicit Fredholm conditions for pseudodifferential operators on non-compact and singular manifolds, using the concept of Fredholm groupoids and their representations, with applications to various geometric settings.

## Contribution

It introduces the concept of Fredholm groupoids and characterizes Fredholm conditions via ellipticity and invertibility of limit operators, extending to complex geometric structures.

## Key findings

- Fredholm operators are characterized by ellipticity and invertibility of limit operators.
- The class of stratified submersion groupoids has Exel's property, ensuring exhaustiveness of regular representations.
- Fredholm conditions apply to manifolds with singularities, desingularizations, and blow-ups.

## Abstract

We give explicit Fredholm conditions for classes of pseudodifferential operators on suitable singular and non-compact spaces. In particular, we include a "user's guide" to Fredholm conditions on particular classes of manifolds including asymptotically hyperbolic manifolds, asymptotically Euclidean (or conic) manifolds, and manifolds with poly-cylindrical ends. The reader interested in applications should be able read right away the results related to those examples, beginning with Section 5. Our general, theoretical results are that an operator adapted to the geometry is Fredholm if, and only if, it is elliptic and all its limit operators, in a sense to be made precise, are invertible. Central to our theoretical results is the concept of a Fredholm groupoid, which is the class of groupoids for which this characterization of the Fredholm condition is valid. We use the notions of exhaustive and strictly spectral families of representations to obtain a general characterization of Fredholm groupoids. In particular, we introduce the class of the so-called groupoids with Exel's property as the groupoids for which the regular representations are exhaustive. We show that the class of "stratified submersion groupoids" has Exel's property, where stratified submersion groupoids are defined by glueing fibered pull-backs of bundles of Lie groups. We prove that a stratified submersion groupoid is Fredholm whenever its isotropy groups are amenable. Many groupoids, and hence many pseudodifferential operators appearing in practice, fit into this framework. This fact is explored to yield Fredholm conditions not only in the above mentioned classes, but also on manifolds that are obtained by desingularization or by blow-up of singular sets.