We study a generalization of the geodesic spray and give conditions for noncomapct manifolds with a Lie structure at infinity to have positive injectivity radius. We also prove that the geometric operators are generated by the given Lie algebra of vector fields. This is the first one in a series of papers devoted to the study of the analysis of geometric differential operators on manifolds with Lie structure at infinity.2000 Mathematics Subject Classification: 53C21, 53C27, 58J40.1. Introduction. Geometric differential operators on complete noncompact Riemannian manifolds were extensively studied due to their applications to physics, geometry, number theory, and numerical analysis. Still, their properties are not as well understood as those of differential operators on compact manifolds, one of the main reasons being that differential operators on noncompact manifolds do not enjoy some of the most useful properties enjoyed by their counterparts on compact manifolds.For example, elliptic operators on noncompact manifolds are not Fredholm in general. (We use the term "elliptic" in the sense that the principal symbol is invertible outside the zero section.) Also, one does not have a completely satisfactory pseudodifferential calculus on an arbitrary complete noncompact Riemannian manifold, which might allow us to decide whether a given geometric differential operator is bounded, Fredholm, or compact (see however [2] and the references therein).However, if one restricts oneself to certain classes of complete noncompact Riemannian manifolds, one has a chance to obtain more precise results on the analysis of the geometric differential operators on those spaces. This paper is the first in a series of papers devoted to the study of such a class of Riemannian manifolds, the class of Riemannian manifolds with a "Lie structure at infinity" (see Definition 3.1). We stress here that few results on the geometry of these manifolds have a parallel in the literature, although there are a fair number of papers devoted to the analysis on particular classes of such manifolds [12,13,15,16,17,38,39,55,57,64,67,70,74,75,76,78] A manifold M 0 with a Lie structure at infinity has, by definition, a natural compactification to a manifold with corners M = M 0 ∪∂M such that the tangent bundle T M 0 → M 0

We define and study an algebra Ψ ∞ 1,0,V (M 0 ) of pseudodifferential operators canonically associated to a noncompact, Riemannian manifold M 0 whose geometry at infinity is described by a Lie algebra of vector fields V on a compactification M of M 0 to a compact manifold with corners. We show that the basic properties of the usual algebra of pseudodifferential operators on a compact manifold extend to Ψ ∞ 1,0,V (M 0 ). We also consider the algebra Diff * V (M 0 ) of differential operators on M 0 generated by V and C ∞ (M ), and show that. Our construction solves a problem posed by Melrose in 1990. Finally, we introduce and study semi-classical and "suspended" versions of the algebra Ψ ∞ 1,0,V (M 0 ).

The first five sections of this paper are a survey of algebras of pseudodifferential operators on groupoids. We thus review differentiable groupoids, the definition of pseudodifferential operators on groupoids, and some of their properties. We use then this background material to establish a few new results on these algebras that are useful for the analysis of geometric operators on non-compact manifolds and singular spaces. The first step is to establish that the geometric operators on groupoids are in our algebras. This then leads to criteria for Fredholmness for geometric operators on suitable non-compact manifolds, as well as to an inductive procedure to study their essential spectrum. As an application, we answer a question of Melrose on the essential spectrum of the Laplace operator on manifolds with multi-cylindrical ends.Contents 35 10. Examples III: Applications 39 References 41 b-calculus. The first author would like to thank R. Melrose, J. Cuntz, and the SFB 478 Geometrische Strukturen in der Mathematik, for their warm hospitality and useful discussions at MIT and, respectively, at the University of Münster, where part of this work was done.

We study the complex powers A z of an elliptic, strictly positive pseudodifferential operator A using an axiomatic method that combines the approaches of Guillemin and Seeley. In particular, we introduce a class of algebras, "extended Weyl algebras," whose definition was inspired by Guillemin's paper [11]. An extended Weyl algebra can be thought of as an algebra of "abstract pseudodifferential operators." Many algebras of pseudodifferential operators are extended Weyl algebras. Several results typical for algebras of pseudodifferential operators (asymptotic completeness, construction of Sobolev spaces, boundedness between apropriate Sobolev spaces, ... ) generalize to extended Weyl algebras. Most important, our results may be used to obtain precise estimates at infinity for A z , when A > 0 is elliptic and defined on a non-compact manifold, provided that a suitable ideal of regularizing operators is specified (a submultiplicative Ψ * -algebra). We shall use these results in a forthcoming paper to study pseudodifferential operators and Sobolev spaces on manifolds with a Lie structure at infinity (a certain class of non-compact manifolds). ∞ k=0 H k (T ) is endowed with the system of norms (q k ) k≥0 . Proposition A.4. For k ≥ 0 ∪ {∞}, the canonical bilinear map J k × H 0 (T ) −→ H k (T ) : (A, f ) −→ Af is well-defined and continuous.Proof. This follows immediately by induction from the definitions.

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