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23-27 juin 2025 Paris (France)
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Research talksChris Bruce - Ample groupoids from rings: rigidity and homology Each ring gives rise to an ample (étale) groupoid. I will present on joint work with Xin Li in which we prove several rigidity theorems for this construction, showing that the groupoid may remember a surprising amount of information about the ring. I will also discuss joint work with Yosuke Kubota and Takuya Takeishi where we compute groupoid homology for the groupoids attached to rings of algebraic integers in number fields. Our homology computations combined with a recent breakthrough by Xin Li lead to simplicity results for the topological full groups of our groupoids. Paulo Carrillo-Rouse - Assembling assembly maps for discrete group For a discrete group G, many interesting invariants (higher indices) live in the K-theory of the group C*-algebra. A way of assembling all these invariants (conjectured) was given by Baum and Connes in the early 80's by assembling all possible G-family indices for appropriate proper actions using pushforward maps. Using deformations groupoids and ideas coming from Hilsum and Skandalis I have been working during the last years the original pushforward model proposed by Baum and Connes (BC), for example, together with Bai-Ling Wang and Hang Wang we were able to compute a complete index pairing formula for the pairing of the BC left hand with the periodic cyclic cohomology of the group. Also, with the use of deformation groupoids it is possible to give a geometric characterization of the rational injectivity of the BC map. After resuming all these previous works I will go further and report on a joint work with my phd student Quentin Karegar, I will explain how the pushforward model is quite natural to compare different assembly maps for different groups, for example for a given group morphism I will explain how one can define natural pushforwards between the BC left hand side groups. In doing so, a new feature arises, one requires to extend the possible cycles to orbifolds cycles, this leads in one side to interesting Orbifold index formulas and on the other hand to interesting questions about the possibility of assembling different assembly maps (I will try to give an appropiate sense to this last phrase). Amandine Escalier - Measure and orbit equivalence of graph products Graph products were introduced by Green in 1990 and generalise constructions such as free or direct products of groups, as well as right angled Artin groups. In this talk we will present a joint work with Camille Horbez, classifying graph products up to measure and orbit equivalence. We will highlight the role of measured groupoids in the latter classification as well as the links with Kida’s work on Mapping Class Groups superrigidity. Vito Felice Zenobi - Delocalized Chern characters through deformation to the normal cone groupoids Let (M, 𝔽) be a compact foliated manifold and let G its monodromy groupoid. We can then define a long exact sequence of K-theory groups of C*-algebras, called the adiabatic exact sequence, which is the receptacle of intex theoretic invariants: the primary ones in the K-theory of the groupoid C*-algebra of G and the secondary invariants in the K-theory group associated to its adiabatic deformation, the so-called ρ-classes.
Let us fix a complete transversal T of (M, 𝔽) and the étale subgroupoid Γ of G obtained by restricion to T, then for any dense and holomorphically closed subalgebra 𝒜(Γ) in C*r(Γ), we are going to define a Chern type mapping from the adiabatic exact sequence to a long exact sequence of periodic homology groups, which in particular sends the ρ-classes to the delocalized preiodic homology group of 𝒜(Γ).
Under suitable hypoteses about the foliation, through pairing with delocalized periodic cocycles, it is possible to extract numeric ρ-invariants, associated to Dirac type operators and obtain geometric and topological applications about (M, 𝔽).
Samuel Mellick - Vanishing rank gradient via groupoid cost In recent work with Mikolaj Fraczyk and Amanda Wilkens, we proved that higher rank semisimple Lie groups (such as SL(3,R)) have fixed price one. An application of this is to show that the minimum number of generators for lattices in such groups is little-o of their covolume, that is, they have vanishing rank gradient. Marco Moraschini - Simplicial volume and complete affine manifolds The topology of affine manifolds is still poorly understood despite a number of classical conjectures: Chern conjecture predicts that all closed affine manifolds have vanishing Euler characteristic, whereas Auslander conjecture predicts that the fundamental group of complete affine manifolds is virtually solvable. In this talk we will discuss the relations of these classical conjectures with a question posed independently by Lück and Bucher—Connell—Lafont on the vanishing of the simplicial volume of aspherical affine manifolds. In particular, we will show a vanishing result for the simplicial volume and its dynamical version of certain complete affine manifolds that answers a question by Lück in this setting. This is a joint work with Alberto Casali. Peter Naryshkin - Quantitative orbit equivalence for Z-odometer Quantitative orbit equivalence is a study of orbit equivalences between two p.m.p. actions of (possibly different) groups, where one additionally asks some integrability condition on the cocycle. There are several results showing that such orbit equivalence (with suitable conditions) must preserve certain invariants of the action and the acting group (e.g., entropy and growth). On the other hand, given two actions with the same invariants, it is a difficult problem to decide whether they are orbit equivalent with a sufficiently good integrability of the cocycle. In the first part of my talk, I will give a brief introduction to the topic and describe what is known. After that, I will sketch a proof of our main result, which says that any two Z-odometers are Lp-orbit equivalent for any p < 1. Based on a joint ongoing work with Spyros Petrakos. Anush Tserunyan - Measure equivalence of Baumslag–Solitar groups In 2001, Whyte proved that all Baumslag–Solitar groups BS(r,s), with |r|,|s|≠1 and |r|≠|s|, are quasi-isometric, thereby completing the quasi-isometry classification of Baumslag–Solitar groups initiated by Farb and Mosher. Since then, the question as to their measure equivalence has remained an intriguing open problem. Together with D. Gaboriau, A. Poulin, R. Tucker-Drob, and K. Wrobel, we solve this problem, establishing the measure equivalence counterpart to Whyte's theorem. Although measure equivalence concerns measure-preserving actions, we reduce it to a measured graph-theoretic problem in the non-measure-preserving (type III) setting. We then solve this graph-theoretic problem using descriptive set-theoretic constructions of measured graphs and a new analysis of the type III_0 setting via the associated Krieger flow. |
