Gluing action groupoids: differential operators and Fredholm conditions

TL;DR

This paper establishes Fredholm criteria for differential operators on open manifolds using groupoid-based algebraic structures, linking ellipticity and invertibility of limit operators on amenable Lie groups.

Contribution

It introduces a novel gluing method for groupoids to analyze differential operators and derives Fredholm conditions in this framework, extending previous results to new classes of manifolds.

Findings

Fredholm operators characterized by ellipticity and invertibility of limit operators

Construction of a groupoid via gluing reductions of action groupoids

Applicable to various classes of open manifolds with amenable Lie group actions

Abstract

We prove some Fredholm conditions for many algebras of differential operators on particular classes of open manifolds, which include asymptotically Euclidean or asymptotically hyperbolic manifolds. Our typical result is that an operator $P$ is Fredholm if, and only if, it is elliptic and some limit operators $(P_\alpha)_{\alpha \in A}$ are invertible. The operators $P_\alpha$ are right-invariant operators on amenable Lie groups $G_\alpha$, and are of the same type of $P$. To obtain this result, we consider algebras of differential operators that are generated by groupoids. We study a general gluing procedure for goupoids, and use it to construct a groupoid $\mathcal{G}$ by gluing reductions of action groupoids $(X_i \rtimes G_i)_{i \in I}$. We show that when each Lie groups $G_i$ is amenable and acts trivially on $\partial X_i$, then the differential operators generated by $\mathcal{G}$ satisfy the aforementionned Fredholm conditions. Many classes of differential operators on open manifolds satisfy these conditions, and we give several examples.