Orbit Equivalence and Measured Group Theory

We give a survey of various recent developments in orbit equivalence and measured group theory. This subject aims at studying infinite countable groups through their measure preserving actions.Comment: 2010 Hyderabad ICM proceeding; Dans Proceedings of the International Congress of Mathematicians, Hyderabad, India - International Congress of Mathematicians (ICM), Hyderabad : India (2010

Relative Property (T) Actions and Trivial Outer Automorphism Groups

We show that every non-amenable free product of groups admits free ergodic probability measure preserving actions which have relative property (T) in the sense of S.-Popa \cite[Def. 4.1]{Pop06}. There are uncountably many such actions up to orbit equivalence and von Neumann equivalence, and they may be chosen to be conjugate to any prescribed action when restricted to the free factors. We exhibit also, for every non-amenable free product of groups, free ergodic probability measure preserving actions whose associated equivalence relation has trivial outer automorphisms group. This gives in particular the first examples of such actions for the free group on $2$ generators.Comment: Improvement in the presentation of the replacement trick. Introduction of compressions for non-ergodic equivalence relation

Invariant Percolation and Harmonic Dirichlet Functions

The main goal of this paper is to answer question 1.10 and settle conjecture 1.11 of Benjamini-Lyons-Schramm [BLS99] relating harmonic Dirichlet functions on a graph to those of the infinite clusters in the uniqueness phase of Bernoulli percolation. We extend the result to more general invariant percolations, including the Random-Cluster model. We prove the existence of the nonuniqueness phase for the Bernoulli percolation (and make some progress for Random-Cluster model) on unimodular transitive locally finite graphs admitting nonconstant harmonic Dirichlet functions. This is done by using the device of $\ell^2$ Betti numbers.Comment: to appear in Geometric And Functional Analysis (GAFA

What is Cost?

Two-page paper on the notion of cost of groups and measured equivalence relations to appear in the "What is?" series in the Notices of the AMS.Comment: 2 page

Compte rendu de : LAHIRE, Bernard. Pour la sociologie. Et pour en finir avec une prétendue « culture de l’excuse ». Paris, La Découverte, 2016. 183 p.

Compte rendu d’un essai de Bernard Lahire, essai écrit en réponse à un pamphlet de Philippe Val.Compte rendu d’un essai écrit en réponse au pamphlet de Philippe Val, Malaise dans l’inculture (Paris, Grasset, 2015) pour qui le « sociologisme » constitue le grand mal actuel de notre société

Approximations of standard equivalence relations and Bernoulli percolation at p\_u

The goal of this note is to announce certain results in orbit equivalence theory, especially concerning the approximation of p.m.p. standard equivalence relations by increasing sequence of sub-relations, with applications to the behavior of the Bernoulli percolation on graphs at the threshold pu. R\'esum\'e en Fran\c{c}ais. Approximations de relations d'\'equivalence standards et percolation de Bernoulli \`a p\_u

Le droit et les marchands du temple

International audienceÀ en croire l'Évangile selon saint Jean, Jésus entra dans le Temple et en chassa les marchands de boeufs, de brebis et de colombes ainsi que les changeurs. Un type de commerces qui ne se trouve plus guère dans les édifices cultuels ! Pour autant, les commerçants n'en sont pas totalement absents, surtout à proximité, le tourisme cultuel justifiant leur installation autour de certains lieux de culte (Notre-Dame de Paris, Lourdes, Rome, Lisieux...). Et si certaines religions furent un temps réticentes à autoriser ces activités mercantiles, elles les considèrent aujourd'hui comme une activité accessoire de la mission cultuelle

Examples of Groups that are Measure Equivalent to the Free Group

Published: Ergodic Theory & Dynam. Systems, 25 (2005), no. 6, 1809-1827.International audienceMeasure Equivalence (ME) is the measure theoretic counterpart of quasi-isometry. This field grew considerably during the last years, developing tools to distinguish between different ME classes of countable groups. On the other hand, contructions of ME equivalent groups are very rare. We present a new method, based on a notion of measurable free-factor, and we apply it to exhibit a new family of groups that are measure equivalent to the free group. We also present a quite extensive survey on results about Measure Equivalence for countable groups