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We prove that all isometric actions of higher rank simple Lie groups and their lattices on arbitrary uniformly convex Banach spaces have a fixed point. This vastly generalises a recent breakthrough of Oppenheim. Combined with earlier work of Lafforgue and of Liao on strong Banach property (T) for non-Archimedean higher rank simple groups, this confirms a long-standing conjecture of Bader, Furman, Gelander and Monod. As a consequence, we deduce that sequences of Cayley graphs of finite quotients of a higher rank lattice are super-expanders., 30 pages ; minor changes in v2

The word stable is used to describe a situation when mathematical objects that almost satisfy an equation are close to objects satisfying it exactly. We study operator-algebraic forms of stability for unitary representations of groups and quantum synchronous strategies for non-local games. We observe in particular that simple spectral gap estimates can lead to strong quantitative forms of stability. For example, we prove that the direct product of two (flexibly) Hilbert-Schmidt stable groups is again (flexibly) Hilbert-Schmidt stable, provided that one of them has Kazhdan's property (T). We also provide a simple form and simple analysis of a non-local game with few questions, with the property that synchronous strategies with large value are close to perfect strategies involving large Pauli matrices. This simplifies one of the steps (the question reduction) in the recent announced resolution of Connes' embedding problem by Ji, Natarajan, Vidick, Wright and Yuen.

We present a simple tool to perform analysis with groups such as SL(n,R) and SL(n,Z), that has been introduced by Vincent Lafforgue in his study of non-unitary representations, in connection with the Baum-Connes conjecture and strong property (T). It has been later applied in various contexts: operator algebras, Fourier analysis, geometry of Banach spaces or dynamics. The idea is to first restrict to compact subgroups and then exploit how they sit inside the whole group., Prepared for the proceedings of the ICM2022 (20 pages)

If $G$ is a group and $S$ a generating set, $G$ canonically embeds into the automorphism group of its Cayley graph and it is natural to try to minimize, over all generating sets, the index of this inclusion. This infimum is called the Cayley index of the group. In a recent series of works, we have characterized the infinite finitely generated groups with Cayley index $1$. We complement this characterization by showing that the Cayley index is $2$ in the remaining cases and is attained for a finite generating set., 9 pages

Given a set $B$ of operators between subspaces of $L_p$ spaces, we characterize the operators between subspaces of $L_p$ spaces that remain bounded on the $X$-valued $L_p$ space for every Banach space on which elements of the original class $B$ are bounded. This is a form of the bipolar theorem for a duality between the class of Banach spaces and the class of operators between subspaces of $L_p$ spaces, essentially introduced by Pisier. The methods we introduce allow us to recover also the other direction --characterizing the bipolar of a set of Banach spaces--, which had been obtained by Hernandez in 1983., Comment: 34 pages. Old project, already announced at several occasions in 2016, and that took a long time to be completed. Comments welcome v2: 37 pages. Section 5 added on the duality between Banach spaces and operators on full Lp spaces. A few references added

We prove that cocompact (and more generally: undistorted) lattices on $\tilde{A}_2$-buildings satisfy Lafforgue's strong property (T), thus exhibiting the first examples that are not related to algebraic groups over local fields. Our methods also give two further results. First, we show that the first $\ell^p$-cohomology of an $\tilde{A}_2$-building vanishes for any finite $p$. Second, we show that the non-commutative $L^p$-space for $p$ not in $[\frac 4 3,4]$ and the reduced $C^*$-algebra associated to an $\tilde{A}_2$-lattice do not have the operator space approximation property and, consequently, that the lattice is not weakly amenable., Comment: v1: 68 pages, 6 figures; v2: 79 pages, many improvements in the presentation. To appear in Ann. Sci. \'Ecole Norm. Sup

We establish precise regularity conditions for $L_p$-boundedness of Fourier multipliers in the group algebra of $SL_n(\mathbf{R})$. Our main result is inspired by H��rmander-Mikhlin criterion from classical harmonic analysis, although it is substantially and necessarily different. Locally, we get sharp growth rates of Lie derivatives around the singularity and nearly optimal regularity order. The asymptotics also match Mikhlin formula for a exponentially growing weight with respect to the word length. Additional decay comes imposed by this growth and Mikhlin condition for high order terms. Lafforgue/de la Salle's rigidity theorem fits here. The proof includes a new relation between Fourier and Schur $L_p$-multipliers for nonamenable groups. By transference, matters are reduced to a rather nontrivial $RC_p$-inequality for $SL_n(\mathbf{R})$-twisted forms of Riesz transforms associated to fractional laplacians. Our second result gives a new and much stronger rigidity theorem for radial multipliers in $SL_n(\mathbf{R})$. More precisely, additional regularity and Mikhlin type conditions are proved to be necessary up to an order $\sim |\frac12 - \frac1p| (n-1)$ for large enough $n$ in terms of $p$. Locally, necessary and sufficient growth rates match up to that order. Asymptotically, extra decay for the symbol and its derivatives imposes more accurate and additional rigidity in a wider range of $L_p$-spaces. This rigidity increases with the rank, so we can construct radial generating functions satisfying our H��rmander-Mikhlin sufficient conditions in rank $n$ and failing the rigidity conditions for ranks $m >> n$. We also prove automatic regularity and rigidity estimates for first and higher order derivatives of $\mathrm{K}$-biinvariant multipliers in the rank 1 groups $SO(n,1)$., To appear in Duke Math. J

La question générale étudiée dans ce mémoire d’habilitation est de comprendre dans quelle mesure certains objets mathématiques associés à un groupe peuvent être déformés ou approchés non trivialement. On parle de malléabilité lorsque c'est possible et de rigidité sinon. En fonctions des chapitres, ces objets seront des représentations sur des espaces de Banach, des algèbres d'opérateurs, des espaces Lp non commutatifs ou des graphes de Cayley. Les trois premiers chapitres traitent de rigidité pour les groupes algébriques de rang supérieur dans un cadre linéaire, et plus précisément dans un cadre d'algèbres d'opérateurs, d'analyse harmonique et de représentations de groupes sur des espaces de Banach. On y démontre que les algèbres d’opérateurs associées aux réseaux en rang supérieur ont de très mauvaises propriétés d’approximation, de plus en plus lorque le rang croît. On prouve aussi une forme renforcée de la propriété (T) pour les actions sur les espaces de Banach. Le quatrième chapitre ne porte que sur l'aspect malléabilité. C'est une présentation des mes travaux sur la moyennabilité des groupes discrets et sur ce qu'on appelle les actions extensivement moyennables. Le cinquième et dernier chapitre porte sur mes travaux sur les aspects de malléabilité et de rigidité pour les graphes de Cayley de groupes de présentation finie, et plus généralement pour les graphes transitifs simplement connexes à grande échelle. J'y étudie l'espace topologique de tous les tels graphes, en essayant de comprendre les points isolés (les graphes ``rigides'') et les points non isolés (les graphes ``malléables'').

La question générale étudiée dans ce mémoire d’habilitation est de comprendre dans quelle mesure certains objets mathématiques associés à un groupe peuvent être déformés ou approchés non trivialement. On parle de malléabilité lorsque c'est possible et de rigidité sinon. En fonctions des chapitres, ces objets seront des représentations sur des espaces de Banach, des algèbres d'opérateurs, des espaces Lp non commutatifs ou des graphes de Cayley. Les trois premiers chapitres traitent de rigidité pour les groupes algébriques de rang supérieur dans un cadre linéaire, et plus précisément dans un cadre d'algèbres d'opérateurs, d'analyse harmonique et de représentations de groupes sur des espaces de Banach. On y démontre que les algèbres d’opérateurs associées aux réseaux en rang supérieur ont de très mauvaises propriétés d’approximation, de plus en plus lorque le rang croît. On prouve aussi une forme renforcée de la propriété (T) pour les actions sur les espaces de Banach. Le quatrième chapitre ne porte que sur l'aspect malléabilité. C'est une présentation des mes travaux sur la moyennabilité des groupes discrets et sur ce qu'on appelle les actions extensivement moyennables. Le cinquième et dernier chapitre porte sur mes travaux sur les aspects de malléabilité et de rigidité pour les graphes de Cayley de groupes de présentation finie, et plus généralement pour les graphes transitifs simplement connexes à grande échelle. J'y étudie l'espace topologique de tous les tels graphes, en essayant de comprendre les points isolés (les graphes ``rigides'') et les points non isolés (les graphes ``malléables'').

26 pages; International audience; We prove that connected higher rank simple Lie groups have Lafforgue's strong property (T) with respect to a certain class of Banach spaces E10 containing many classical superreflexive spaces and some non-reflexive spaces as well. This generalizes the result of Lafforgue asserting that SL(3,R) has strong property (T) with respect to Hilbert spaces and the more recent result of the second named author asserting that SL(3,R) has strong property (T) with respect to a certain larger class of Banach spaces. For the generalization to higher rank groups, it is sufficient to prove strong property (T) for Sp(2,R) and its universal covering group. As consequences of our main result, it follows that for the Banach spaces X∈E10, connected higher rank simple Lie groups and their lattices have property (FX) of Bader, Furman, Gelander and Monod, and that the expanders contructed from a lattice in a connected higher rank simple Lie group do not admit a coarse embedding into X.

In this short note we give a short proof of a recent result by Potapov and Sukochev (arXiv:0904.4095v1), stating that a Lipschitz function on the real line remains Lipschitz on the (self-adjoint part of) non-commutative $L_p$ spaces with $1, Comment: 3 pages

We characterize the finitely generated groups that admit a Cayley graph whose only automorphisms are the translations, confirming a conjecture by Watkins from 1976. The proof relies on random walk techniques. As a consequence, every finitely generated group admits a Cayley graph with countable automorphism group. We also treat the case of directed graphs., v1: 17 pages v2: 19 pages, improvements in the presentation, Section 6 added. To appear in Annales Henri Lebesgue

��uk proved that if a finitely generated group admits a Cayley graph such that the Laplacian on the links of this Cayley graph has a spectral gap $> \frac{1}{2}$, then the group has property (T), or equivalently, every affine isometric action of the group on a Hilbert space has a fixed point. We prove that the same holds for affine isometric actions of the group on a uniformly curved Banach space (for example an $L^p$-space with $1 < p < \infty$ or an interpolation space between a Hilbert space and an arbitrary Banach space) as soon as the Laplacian on the links has a two-sided spectral gap $>1-\varepsilon$. This criterion applies to random groups in the triangular density model for densities $> \frac{1}{3}$. In this way, we are able to generalize recent results of Dru��u and Mackay to affine isometric actions of random groups on uniformly curved Banach spaces. Also, in the setting of actions on $L^p$-spaces, our results are quantitatively stronger, even in the case $p=2$. This naturally leads to new estimates on the conformal dimension of the boundary of random groups in the triangular model. Additionally, we obtain results on the eigenvalues of the $p$-Laplacian on graphs, and on the spectrum and degree distribution of Erd��s-R��nyi graphs., v2: Major revision; substantial improvements of the main results; 28 pages v3: Minor revision; 29 pages

A vertex-transitive graph X is called local-to-global rigid if there exists R such that every other graph whose balls of radius R are isometric to the balls of radius R in X is covered by X. Let $d\geq 4$. We show that the 1-skeleton of an affine Bruhat-Tits building of type $\widetilde A_{d-1}$ is local-to-global rigid if and only if the underlying field has characteristic 0. For example the Bruhat-Tits building of $SL(d,F_p((t)))$ is not local-to-global rigid, while the Bruhat-Tits building of $SL(d,Q_p)$ is local-to-global rigid., 11 pages

We prove some noncommutative analogues of a theorem by Plotkin and Rudin about isometries between subspaces of Lp-spaces. Let 0, Comment: 30 pages; revised version of the paper with previous title "Equimeasurabily and isometries in noncommutative Lp-spaces". Changes in the title, presentation and content. Added results on unbounded operators, and applications

If $E$ is an operator space, the non-commutative vector valued $L^p$ spaces $S^p[E]$ have been defined by Pisier for any $1 \leq p \leq \infty$. In this paper a necessary and sufficient condition for a Hankel matrix of the form $(a_{i+j})_{0 \le i,j}$ with $a_k \in E$ to be bounded in $S^p[E]$ is established. This extends previous results of Peller where $E=\C$ or $E=S^p$. The main theorem states that if $1 \leq p < \infty$, $(a_{i+j})_{0 \le i,j}$ is bounded in $S^p[E]$ if and only if there is an analytic function $\phi$ in the vector valued Besov Space $B_p^{1/p}(E)$ such that $a_n = \hat \phi(n)$ for all $n \in \N$. In particular this condition only depends on the Banach space structure of $E$. We also show that the norm of the isomorphism $\phi \mapsto (\hat \phi(i+j))_{i,j}$ grows as $\sqrt p$ as $p \to \infty$, and compute the norm of the natural projection onto the space of Hankel matrices., Comment: 17 pages