Gluing action groupoids: differential operators and Fredholm conditions

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.