Minicourse

I. Introduction to Superrigidity
II. Orbit Equivalence in Ergodic Theory
III. Measurable Group theory
IV. Actions on manifolds

The general theme of these lectures is the "rigidity" that a structure of a group can impose on its actions in various settings: linear representations, actions on manifolds, orbit structures of measurable actions etc. Our goal is to describe the general landscape of the questions, results and (some of the) ideas which appear in this area.

Titles and abstracts of talks:

Paul Baum: Cosheaf homology
Given a local system on a topological space X (i.e. a representation of the fundamental groupoid of X) both the homology and the cohomology of X with coefficients in the local system are defined. This talk considers what happens when the defining conditions for a local system are relaxed to obtain sheaves and cosheaves. With sheaves one can do cohomology and with cosheaves homology. Cosheaves arise naturally from group actions on topolgical spaces, and enter into Chern character problems relevant to the Baum-Connes conjecture. Recent results of C. Voigt relate cosheaf homology to Bredon homology and to periodic cyclic homology.

Moulay-Tahar Benameur: Homotopy invariance of Higher signatures in Haefliger cohomology
We shall explain in this talk some recent results obtained in collaboration with James Heitsch and related with index theory for foliations. We shall focus on the definition of the leafwise signature in Haefliger cohomology and sketch a new geometric proof of the leafwise homotopy invariance of higher signatures with coefficients in leafwise flat bundles.

Paulo Carrillo: Analytical indices for Lie groupoids
For a Lie groupoid there is an analytic index morphism taking values in the K-theory of the C*-algebra associated to the groupoid. This is a good invariant, but extracting numerical invariants from it, with the existent tools, is very difficult. In this talk, we will explain how to define another analytic index morphisms associated to the Lie groupoid. These ones take values in some groups that allow us to do pairings with cyclic cocycles. We obtain some abstract index formulas.

Claire Debord: Poincaré Duality for stratified pseudomanifolds
We associate to a stratified pseudomanifold X a differentiable groupoid T^{S}X which plays the role of the tangent space to X. We construct a Dirac element. Thanks to a recursive process on the depth of the stratification together with the stability of the constructions which are involved we show that the Dirac element induces a K-duality between the C*-algebras C*(T^{S}X) and C(X).

Alexander Gorokhovsky: Deformation quantization of gerbes on etale groupoids
This is a joint work with P.Bressler, R. Nest and B. Tsygan. We will discuss problems in which formal deformations of etale groupoids and gerbes on them arise and give an explicit description of the differential graded Lie algebra which controls this deformation theory. Deformation quantization of gerbes on etale groupoids,

Eli Hawkins: A groupoid approach to quantization
I define a notion of "polarization" for Lie groupoids. Using polarized symplectic groupoids, I present a general strategy for constructing C*-algebras that quantize Poisson manifolds. This unifies previous constructions including classical geometric quantization of a symplectic manifold and the C*-algebra of a source- connected Lie groupoid.

Adrian Ioana: Orbit inequivalent actions for groups containing a copy of F_2
I will prove that any countable discrete group G which contains a copy F_2 admits uncountably many non orbit equivalent actions.

Steven Hurder: Index theory and LS category for Riemannian foliations

Alex Kumjian: Fell bundles associated to groupoid morphisms
Given a continuous open surjective morphism $\pi :G \to H$ of \'etale groupoids with amenable kernel, we construct a Fell bundle $E$ over $H$ and prove that its C*-algebra $C^*_r(E)$ is isomorphic to $C^*_r(G)$. This is related to results of Fell concerning C*-algebraic bundles over groups. The case $H=X$, a locally compact space, was treated by Ramazan. We conclude that $C^*_r(G)$ is strongly Morita equivalent to a crossed product, the C*-algebra of a Fell bundle arising from an action of the groupoid $H$ on a C*-bundle over $H^0$. We apply the theory to groupoid morphisms obtained from extensions of dynamical systems and from morphisms of directed graphs with the path lifting property. This is joint work with Valentin Deaconu and Birant Ramazan.

Klaas Landsman and Rogier Bos: Continuous representations of Lie groupoids
The notion of a representation of a locally compact groupoid G with Haar system on a measurable field of Hilbert spaces has been developed by Jean Renault and has the advantage of yielding a connection with the representation theory of the associated C*- algebra C*(G). However, in the case of Lie groupoids it is worth studying representations on continuous fields of C*-algebras (i.e. on Hilbert C*-modules over C_0(M), where M is the base space of G). Using ideas from geometric quantization, we present a method to construct such representations, as well as an associated "quantization commutes with reduction" theorem in case that G is proper.

Jean-Marie Lescure: An index theorem for conical pseudomanifolds
We define an analytical index map and a topological index map for conical pseudomanifolds. These constructions generalize the analogous constructions used by Atiyah and Singer in the proof of their index theorem for a smooth, compact manifold M. A main ingredient is a non-commutative algebra that plays in our setting the role of C_0(T*M). We prove a Thom isomorphism between non-commutative algebras which gives a new example of wrong way functoriality in K-theory. We then give a new proof of the Atiyah-Singer index theorem using deformation groupoids and show how it generalizes to conical pseudomanifolds. We thus prove a topological index theorem for conical pseudomanifolds.

Ieke Moerdijk: Subgroupoids of Lie groupoids

Bertrand Monthubert: Boutet de Monvel's Calculus and Groupoids
Can Boutet de Monvel's algebra on a compact manifold with boundary be obtained as the algebra Psi^0(G) of pseudodifferential operators on some Lie groupoid G? If it could, the kernel {\mathcal G} of the principal symbol homomorphism would be isomorphic to the groupoid C*-algebra C*(G). While the answer to the above question remains open, we exhibit a groupoid G such that C*(G) possesses an ideal I isomorphic to {\mathcal G}. In fact, we prove first that {\mathcal G}\simeq\Psi\otimes{\mathcal K} with the C*-algebra \Psi generated by the zero order pseudodifferential operators on the boundary and the algebra $\mathcal K$ of compact operators. As both \Psi\otimes {\mathcal K} and I are extensions of C(S*Y)\otimes {\mathcal{K}} by {\mathcal{K}} (S*Y is the co-sphere bundle over the boundary) we infer from a theorem by Voiculescu that both are isomorphic.

Victor Nistor: A topological index theorem for manifolds with corners
We show that the analytic and topological index for the groupoid G(M) associated to a compact manifolds with corners coincide. When the faces of M are contractible, these indices are isomorphisms from K^*(TM) to K_*(G(M)) := K_*(C^*G(M)). This is joint work with Bertrand Monthubert.

Alan L. T. Paterson: The E-theoretic descent functor for groupoids
The descent functor enables us to go from equivariant asymptotic morphisms to asymptotic morphisms of crossed product C*-algebras. It is important in a number of contexts, in particular for the Baum-Connes conjecture and the topological index. The functor for locally compact groups was established in their memoir by Guentner, Higson and Trout, and the talk will discuss what can be said in the groupoid case. (Earlier work on this was done by R. Popescu in his thesis.) There seem to be technical difficulties with establishing in complete generality the descent functor for groupoids, but we prove its existence under certain conditions which apply in a number of cases that arise in practice.

Mikael Pichot: The space of triangle buildings

Paolo Piazza: Foliated rho-invariants
I will present some ongoing work in collaboration with Moulay Benameur about the definition of a foliated rho invariant and the proof of some of its stability properties. This invariant generalizes to measured foliations the classical Cheeger-Gromov rho-invariant on Galois coverings. I will first recall work of Keswani and Piazza-Schick dealing with the homotopy invariance of the Cheeger-Gromov rho invariant on Galois coverings under a Baum-Connes assumption on the maximal group C*-algebra of the covering group; I will then move on and explain how these results can be generalized to measured foliations, assuming the bijectivity of the Baum-Connes map for foliations. I shall also explain how index theoretic considerations play a crucial role throughout.

Sorin Popa: On the superrigidity of group actions in ergodic theory and von Neumann algebras
I will explain how the unlikely combination of deformation and rigidity properties of a measure preserving action of a group on a probability space can make it recognizable by merely knowing the orbit equivalence class of the action, or just its von Neumann algebra.

Ian Putnam: A homology theory and C*-algebras for chaotic dynamical systems
Smale spaces were defined by David Ruelle as an abstract approach to the chaotic dynamical systems in Smale's program. He also described how operator algebras may be constructed from such systems. We will describe a kind of homology theory for these systems and how it may be used to compute the K-theory of the C*-algebras.

Anton Savin: Homotopy classification of elliptic operators on manifolds with corners
This is joint work with V.E. Nazaikinskii and B.Yu. Sternin. We compute the group of stable homotopy classes of elliptic operators on manifolds with corners. It turns out that this group is isomorphic to the analytic K-homology group of a certain explicitly constructed C*-algebra.

Stefaan Vaes: Computations of automorphisms and finite index subfactors of certain II_1 factors
We present the first concrete examples of II_1 factors without non-trivial finite index subfactors. We also present relatively easy examples of II_1 factors with trivial outer automorphism group.

Stéphane Vassout: Uniform non amenability and the first l^2 Betti number
In this talk, I will expose a recent work in collaboration with Mikael Pichot. We define uniform Cheeeger isoperimetric constant for a given finitely generated r-discrete measured groupoid of type II_1, and show that it is bounded below by the first Betti number. In particular we derive an invariant for equivalence relations and define a notion of ergodic uniform non amenability for finitely generated groups which is weaker than the usual one.

Funding

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