Smooth Groupoid Research Articles

Tangent maps and tangent groupoid for Carnot manifolds

This paper studies the infinitesimal structure of Carnot manifolds. By a Carnot manifold we mean a manifold together with a subbundle filtration of its tangent bundle which is compatible with the Lie bracket of vector fields. We introduce a notion of differential, called Carnot differential, for Carnot manifolds maps (i.e., maps that are compatible with the Carnot manifold structure). This differential is obtained as a group map between the corresponding tangent groups. We prove that, at every point, a Carnot manifold map is osculated in a very precise way by its Carnot differential at the point. We also show that, in the case of maps between nilpotent graded groups, the Carnot differential is given by the Pansu derivative. Therefore, the Carnot differential is the natural generalization of the Pansu derivative to maps between general Carnot manifolds. Another main result is a construction of an analogue for Carnot manifolds of Connes' tangent groupoid. Given any Carnot manifold (M,H) we get a smooth groupoid that encodes the smooth deformation of the pair M×M to the tangent group bundle GM. This shows that, at every point, the tangent group is the tangent space in a true differential-geometric fashion. Moreover, the very fact that we have a groupoid accounts for the group structure of the tangent group. Incidentally, this answers a well-known question of Bellaïche [11].

The transverse index theorem for proper cocompact actions of Lie groupoids

Given a proper, cocompact action of a Lie groupoid, we define a higher index pairing between invariant elliptic differential operators and smooth groupoid cohomology classes. We prove a cohomological index formula for this pairing by applying the van Est map and algebraic index theory. Finally we discuss in examples the meaning of the index pairing and our index formula.

Pseudodifferential extensions and adiabatic deformation of smooth groupoid actions

The adiabatic groupoid Gad of a smooth groupoid G is a deformation relating G with its algebroid. In a previous work, we constructed a natural action of R on the C*-algebra of zero order pseudodifferential operators on G and identified the crossed product with a natural ideal J(G) of C⁎(Gad). In the present paper we show that C⁎(Gad) itself is a pseudodifferential extension of this crossed product in a sense introduced by Saad Baaj. Let us point out that we prove our results in a slightly more general situation: the smooth groupoid G is assumed to act on a C*-algebra A. We construct in this generalized setting the extension of order 0 pseudodifferential operators Ψ(A,G) of the associated crossed product A⋊G. We show that R acts naturally on Ψ(A,G) and identify the crossed product of A by the action of the adiabatic groupoid Gad with an extension of the crossed product Ψ(A,G)⋊R. Note that our construction of Ψ(A,G) unifies the ones of Connes (case A=C) and of Baaj (G is a Lie group).

Exponential coordinates and regularity of groupoid heat kernels

Abstract We prove that on an asymptotically Euclidean boundary groupoid, the heat kernel of the Laplacian is a smooth groupoid pseudo-differential operator.

Differential Groupoids and Their Application to the Theory of Spacetime Singularities

The transformation groupoid Γ = $$\overline {OM} $$ × G, where $$\overline {OM} $$ is the total space of the generalized frame G-bundle over spacetime with a singular boundary, is not a Lie groupoid but a differential groupoid, i.e., a smooth groupoid in the category of structured spaces. We define this concept and use it to investigate spacetimes with various kinds of singularities. Any differential transformation groupoid can be represented by an algebra of operators on a bundle of Hilbert spaces defined on the groupoid fibers. This algebra reflects the structure of a given fiber even if it is a fiber over a singularity. It is also shown that any spacetime with singularities can be regarded as a noncommutative space. Its geometry is done in terms of a noncommutative algebra defined on the corresponding differential transformation groupoid. We focus on the structure of “malicious singularities” such as the ones appearing in the beginning and in the end of the closed Friedman universe.

A Riemann-Roch Theorem¶for One-Dimensional Complex Groupoids

We consider a smooth groupoid of the form \Sigma\rtimes\Gamma where \Sigma is a Riemann surface and \Gamma a discrete pseudogroup acting on \Sigma by local conformal diffeomorphisms. After defining a K-cycle on the crossed product C_0(\Sigma)\rtimes\Gamma generalising the classical Dolbeault complex, we compute its Chern character in cyclic cohomology, using the index theorem of Connes and Moscovici. This involves in particular a generalisation of the Euler class constructed from the modular automorphism group of the von Neumann algebra L^{\infty}(\Sigma)\rtimes\Gamma.