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We consider the isometry group of the infinite dimensional separable hyperbolic space with its Polish topology. This topology is given by the pointwise convergence. For non-locally compact Polish groups, some striking phenomena like automatic continuity or extreme amenability may happen. Our leading idea is to compare this topological group with usual Lie groups on one side and with non-Archimedean infinite dimensional groups like $\mathcal{S}_\infty$, the group of all permutations of a countable set on the other side. Our main results are Automatic continuity (any homomorphism to a separable group is continuous), minimality of the Polish topology, identification of its universal Furstenberg boundary as the closed unit ball of a separable Hilbert space with its weak topology, identification of its universal minimal flow as the completion of some suspension of the action of the additive group of the reals on its universal minimal flow. All along the text, we lead a parallel study with the sibling group of isometries of a separable Hilbert space., After a first version of this paper, Todor Tsankov asked if the topology is minimal. A positive answer has been added to this second version

We show that a partition of the unity (or POVM) on a Hilbert space that is almost orthogonal is close to an orthogonal POVM in the same von Neumann algebra. This generalizes to infinite dimension previous results in matrix algebras by Kempe-Vidick and Ji-Natarajan-Vidick-Wright-Yuen. Quantitatively, our result are also finer, as we obtain a linear dependance, which is optimal. We also generalize to infinite dimension a duality result between POVMs and minimal majorants of finite subsets in the predual of a von Neumann algebra., Comment: 14 pages; v2: improvements in the presentation and small mistakes corrected. To appear in the open access journal Comptes Rendus - Math\'ematique

Consider a Cremona group endowed with the Euclidean topology introduced by Blanc and Furter. It makes it a Hausdorff topological group that is not locally compact nor metrisable. We show that any sequence of elements of the Cremona group of bounded order that converges to the identity is constant. We use this result to show that the Cremona groups do not contain any non-trivial sequence of subgroups converging to the identity. We also show that, in general, paths in a Cremona group do not lift and do not satisfy a property similar to the definition of morphisms to a Cremona group.

In this work, we investigate the analysis of generators for dynamic graphs, which are defined as graphs whose topology changes over time. We focus on generated graphs whose orders are neither growing nor constant along time. We introduce a novel concept, called "sustainability," to qualify the long-term evolution of dynamic graphs. A dynamic graph is considered sustainable if its evolution does not result in a static, empty, or periodic graph. To measure the dynamics of the sets of vertices and edges, we propose a metric, named "Nervousness," which is derived from the Jaccard distance. As an illustration of how the analysis can be conducted, we design a parametrized generator, named D3G3 (Degree-Driven Dynamic Geometric Graphs Generator), that generates dynamic graph instances from an initial geometric graph. The evolution of these instances is driven by two rules that operate on the vertices based on their degree. By varying the parameters of the generator, different properties of the dynamic graphs can be produced. Our results show that in order to ascertain the sustainability of the generated dynamic graphs, it is necessary to study both the evolution of the order and the Nervousness for a given set of parameters., LIPIcs, Vol. 257, 2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023), pages 8:1-8:19

The Johnson kernel is the subgroup of the mapping class group of a closed oriented surface that is generated by Dehn twists along separating simple closed curves. The rational abelianization of the Johnson kernel has been computed by Dimca, Hain and Papadima, and a more explicit form was subsequently provided by Morita, Sakasai and Suzuki. Based on these results, Nozaki, Sato and Suzuki used the theory of finite-type invariants of 3-manifolds to prove that the torsion subgroup of the abelianized Johnson kernel is non-trivial. In this paper, we give a purely 2-dimensional proof of the non-triviality of this torsion subgroup and provide a lower bound for its cardinality. Our main tool is the action of the mapping class group on the Malcev Lie algebra of the fundamental group of the surface. Using the same infinitesimal techniques, we also provide an alternative diagrammatic description of the rational abelianized Johnson kernel, and we include in the results the case of an oriented surface with one boundary component., 31 pages. Typos corrected in this second version

Let $n\ge 3$, and let $\mathrm{Out}(W_n)$ be the outer automorphism group of a free Coxeter group $W_n$ of rank $n$. We study the growth of the dimension of the homology groups (with coefficients in any field $\mathbb{K}$) along Farber sequences of finite-index subgroups of $\mathrm{Out}(W_n)$. We show that, in all degrees up to $\lfloor\frac{n}{2}\rfloor-1$, these Betti numbers grow sublinearly in the index of the subgroup. When $\mathbb{K}=\mathbb{Q}$, through L\"uck's approximation theorem, this implies that all $\ell^2$-Betti numbers of $\mathrm{Out}(W_n)$ vanish up to degree $\lfloor\frac{n}{2}\rfloor-1$. In contrast, in top dimension equal to $n-2$, an argument of Gaboriau and No\^us implies that the $\ell^2$-Betti number does not vanish. We also prove that the torsion growth of the integral homology is sublinear. Our proof of these results relies on a recent method introduced by Ab\'ert, Bergeron, Fr\k{a}czyk and Gaboriau. A key ingredient is to show that a version of the complex of partial bases of $W_n$ has the homotopy type of a bouquet of spheres of dimension $\lfloor\frac{n}{2}\rfloor-1$.

We classify birational involutions of the real projective plane up to conjugation. In contrast with an analogous classification over the complex numbers (due to E. Bertini, G. Castelnuovo, F. Enriques, L. Bayle and A. Beauville), which includes 4 different classes of involutions, we discover 12 different classes over the reals, and provide many examples when the fixed curve of an involution does not determine its conjugacy class in the real plane Cremona group.

This PhD thesis lies at the interface between topological dynamics and measurable dynamics. First, I study the notion of allosteric actions. These actions are generically free in the sense of the topology but not generically free in the sense of the measure. This surprising behavior highlights the differences between invariant random subgroups and uniformly recurrent subgroups. The nascent theory of quantitative orbit equivalence is the second topic of this thesis. This is a strengthening of orbit equivalence, which aims to understand how metric structures on the orbits of the actions can be distorted. A large part of my work gravitates around one of the founding result of this theory: Belinskaya's theorem., Cette thèse se situe à l'interface entre dynamique topologique et dynamique mesurée. Premièrement, j'y étudie la notion d'action allostérique. Ce sont des actions génériquement libres au sens topologique mais pas génériquement libres au sens de la mesure. Ce comportement étonnant met en valeur les nuances entre sous-groupes aléatoires invariants et sous-groupes uniformément récurrents. Un second sujet d'étude est l'équivalence orbitale quantitative, qui renforce l'équivalence orbitale. Il s'agit de comprendre comment les structures métriques sur les orbites des actions peuvent être distordues par équivalence orbitale. Une grande partie des travaux de cette thèse gravite autour d'un des théorèmes fondateurs de cette théorie : le théorème de Belinskaya.

We define fully irreducible automorphisms of generalized Baumslag-Solitar groups in analogy with fully irreducible automorphisms of free groups. We first obtain a characterization of fully irreducible automorphisms analogous to a condition given by Kapovich. Next we discuss the existence of pseudo-periodic conjugacy classes based on the study of Nielsen paths in a train track representative. Finally we give an algorithm which finds out whether an automorphism with a train track representative is pseudo-atoroidal and fully irreducible., 46 pages

We study lattice embeddings for the class of countable groups $\Gamma$ defined by the property that the largest amenable uniformly recurrent subgroup $A_\Gamma$ is continuous. When $A_\Gamma$ comes from an extremely proximal action and the envelope of $A_\Gamma$ is co-amenable in $\Gamma$, we obtain restrictions on the locally compact groups $G$ that contain a copy of $\Gamma$ as a lattice, notably regarding normal subgroups of $G$, product decompositions of $G$, and more generally dense mappings from $G$ to a product of locally compact groups., Comment: v1: 44 pages, preliminary version. v2: slightly modified version. v3: modified terminology, added paragraph 6.5.4. v4: Part of Section 6 has been extracted to arXiv:2001.08689

International audience; A finite group with an integer representation has a multiplicative action on the ring of Laurent polynomials, which is induced by a nonlinear action on the complex torus. We study the structure of the associated orbit space as the image of the fundamental invariants. For the Weyl groups of types A, B, C and D, this image is a compact basic semi-algebraic set and we present the defining polynomial inequalities explicitly. We show how orbits correspond to solutions in the complex torus of symmetric polynomial systems and give a characterization of the orbit space as the positivity-locus of a symmetric real matrix polynomial. The resulting domain is the region of orthogonality for a family of generalized Chebyshev polynomials, which have connections to topics such as Fourier analysis and representations of Lie algebras.

Given an Artin group $A_\Gamma$, a common strategy in the study of $A_\Gamma$ is the reduction to parabolic subgroups whose defining graphs have small diameter, i.e. showing that $A_\Gamma$ has a specific property if and only if all "small" parabolic subgroups of $A_\Gamma$ have this property. Since "small" parabolic subgroups are the puzzle pieces of $A_\Gamma$ one needs to study their behavior, in particular their intersections. The conjecture we address here says that the class of parabolic subgroups of $A_\Gamma$ is closed under intersection. Under the assumption that intersections of parabolic subgroups in complete Artin groups are parabolic, we show that the intersection of a complete parabolic subgroup with an arbitrary parabolic subgroup is parabolic. Further, we connect the intersection behavior of complete parabolic subgroups of $A_\Gamma$ to fixed point properties and to automatic continuity of $A_\Gamma$ using Bass-Serre theory and a generalization of the Deligne complex., Comment: 19 pages

This thesis is devoted to the study of certain groups of finite Morley rank and odd type. In the first chapter, we consider the problem of the classification of ranked sharply 2-transitive groups from the algebraic point of view of near-domains and near-fields. We study ranked near-fields of characteristic equal to two and under a rather strong technical hypothesis, we prove that they are algebraically closed fields. A characterization of the near-domains in terms of the properties of the half-embedding of the additive K-loop is also sketched. In a second chapter, we establish a number of results on the model theory of uniquely 2-divisible omrga-stable K-loops. The link with the notion of "symétron" introduced by B. Poizat is exploited. We give in this context an analogue of Lascar’s analysis of groups of finite Morley rank. In a third chapter, the class of quasi-Frobenius groups - which contains in particular SO(3, R), PGL(2, C) and GA(1, C) - will be in focus. The question of the identification of these classical groups in a context that extends ranked universes and o-minimal structures is discussed in detail.In a fourth and last chapter, we examine the link between the classification project of ranked quasi-Frobenius groups and definably linear groups over a ranked field. A characterization of definably linear bad pairs in zero characteristic is proposed. We also consider the problem of the solvability of split quasi-Frobenius groups of odd type., Cette thèse est consacrée à l’étude de certains groupes de rang de Morley fini de type impair. Dans un premier chapitre, nous considérons le problème de la classification des groupes strictement 2-transitifs rangés du point de vue algébrique des presque-domaines et des presque-corps. Nous étudions les presque-corps rangés de caractéristique égale à deux et sous une hypothèse technique assez forte, nous démontrons que ce sont des corps algébriquement clos. Une caractérisation des presque-domaines en termes des propriétés du demi-plongement de la K-boucle additive est également esquissée. Dans un deuxième chapitre, nous établissons un certain nombre de résultats sur la théorie des modèles des K-boucles uniquement 2-divisibles omega-stables. Le lien avec la notion de symétron introduite par B. Poizat est exploité. Nous donnons dans ce contexte un analogue de l’analyse de Lascar des groupes de rang de Morley fini. Dans un troisième chapitre, la classe des quasi-groupes de Frobenius - qui contient notamment SO(3, R), PGL(2, C) et GA(1, C) - sera l’objet de notre attention. La question de l’identification de ces groupes classiques dans un contexte qui étend celui des univers rangés et des structures o-minimales est discutée en détail. Dans un quatrième et dernier chapitre, nous examinons le lien entre le projet de classification des quasi-groupes de Frobenius rangés et les groupes définissablement linéaires sur un corps rangé. Une caractérisation des mauvaises paires rangées définissablement linéaires en caractéristique nulle est proposée. On considère également le problème de la résolubilité des quasi-groupes de Frobenius scindés de type impair.

In this thesis we study rigidity properties of automorphism groups of some Gromov hyperbolic groups. In the first part of the thesis, the main group of interest is the group Out(Wn) of outer automorphisms of a universal Coxeter group of rank n, which is the free product of n copies of a cyclic group of order 2. We prove that every isomorphism between finite index subgroups of Out(Wn) is induced by a global conjugation by an element of Out(Wn). In the second part of the thesis, we study the group Out(Fn) of outer automorphisms of a nonabelian free group of rank n. Let H be a subgroup of Out(Fn). We study the existence of generic elements of H in the following sense. We say that an element f of H is generic if, for every conjugacy class c of elements of Fn, we have the following property: there exists a polynomial P with coefficients in ℝ such that the length of fⁿ(c) is equivalent to P(n) if and only if for every element h of H, there exists a polynomial Qh with coefficients in ℝ such that the length of hⁿ(c) is equivalent to Qh(n)., Dans cette thèse, nous étudions des résultats de rigidité de groupes d'automorphismes de certains groupes hyperboliques au sens de Gromov. La première partie de la thèse est consacrée à l'étude du groupe Out(Wn) des automorphismes extérieurs d'un groupe de Coxeter universel de rang n, produit libre de n copies d'un groupe cyclique d'ordre 2. Nous montrons que tout isomorphisme entre sous-groupes d'indices fini de Out(Wn) est induit par une conjugaison globale par un élément de Out(Wn). Dans une deuxième partie, nous étudions le groupe Out(Fn) des automorphismes extérieurs d'un groupe libre non abélien de rang n. Soit H un sous-groupe de Out(Fn). Nous étudions l'existence d'éléments génériques dans H au sens suivant. Un élément f de H est dit générique si, pour toute classe de conjugaison c d'éléments de Fn, nous avons la propriété suivante : il existe un polynôme P à coefficients dans ℝ tel que la longueur de fⁿ(c) est équivalente à P(n) si, et seulement si, pour pour tout élément h de H, il existe un polynôme Qh à coefficients dans ℝ tel que la longueur de hⁿ(c) est équivalente à Qh(n).

We say that two groups are orbit equivalent if they both admit an action on a same probability space that share the same orbits. In particular the Ornstein-Weiss theorem implies that all infinite amenable groups are orbit equivalent to the group of integers. To refine this notion Delabie, Koivisto, Le Maître and Tessera introduced a quantitative version of orbit equivalence and of its measure theoretic counterpart called measure equivalence. They furthermore obtained an upper bound to the possible quantification between two given groups. In this article we offer to answer the inverse problem (find a group being orbit or measure equivalent to a prescribed group with prescribed quantification) in the case of the group of integers, the lamplighter group and Brieussel-Zheng's diagonal products. These results moreover show that the upper bound given by Delabie et al. is optimal: the orbit and measure equivalences we obtain all realize the aforementioned upper bound., On dit que deux groupes sont orbitalement équivalents si tous deux admettent une action sur un même espace de probabilité qui partagent les mêmes orbites (à ensemble de mesure nulle près). Notamment, le théorème d’Ornstein et Weiss stipule que tout groupe infini moyennable est orbitalement équivalent au groupe des entiers. Delabie, Koivisto, Le Maître et Tessera ont introduit une version quantitative de l’équivalence orbitale et de son pendant mesuré afin d’affiner cette relation au sein des groupes moyennables infinis. Ils obtiennent en outre une majoration à la quantification possible entre deux groupes donnés. Dans cet article nous proposons de répondre au problème inverse de la quantification (trouver un groupe qui est orbitalement ou mesure équivalent à un groupe prescrit avec quantification prescrite) dans le cas du groupe des entiers, du groupe d’allumeur de réverbère et d’un produit diagonal de Brieussel-Zheng. Ces résultats montrent de plus que la borne donnée par Delabie et al. est optimale : nos équivalences orbitale et mesurée réalisent la borne en question.

We determine a necessary and sucient condition for the irreducibility of the n-dimensional local homogeneous representation ϕ 3 of the braid group B n for all n ≥ 3. Then, we study the irreducibility of all local linear representations of the braid group B 3 of degree 3. We prove that a local representation of type 1 is reduced to a representation of Burau type. Also, we nd a necessary and sucient condition for the irreducibility of any local representation of type 2. Moreover, we construct a representation of B 3 of degree 6 using the tensor product of local representations of type 2. We determine a necessary and sucient condition that guarantees the irreducibility of the obtained representation.

The Weyl group of a crystallographic root system has a multiplicative action on the ring of multivariate trigonometric polynomials. We study the problem of minimizing an invariant trigonometric polynomial. This problem can be written as a polynomial optimization problem on a compact basic semi-algebraic set. Lasserre's moment-SOS hierarchy is formulated in the basis of generalized Chebyshev polynomials, leading to a different notion of the polynomial degree and smaller matrices in the arising semi-definite program. Bilevel optimization techniques are applied to solve max-min problems. Optimal values of trigonometric polynomials appear in spectral bounds for chromatic numbers and independence numbers of geometric distance graphs. We study the quality of these bounds for polytope norms and prove sharpness in several cases.

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.

This thesis studies the problem of optimizing a trigonometric polynomial with crystallographic symmetry. Optimization of trigonometric polynomials has been subject to many recent works, but a theory for exploiting symmetries has hardly been developed or generalized. We consider the action of a crystallographic group, also known as a Weyl group, on the exponents. This action admits a definition of generalized Chebyshev polynomials. If a trigonometric polynomial is invariant under the action, then it can be written as a sum of generalized Chebyshev polynomial in fundamental invariants.By rewriting an invariant trigonometric polynomial as a classical polynomial, the region of optimization is transformed into the orbit space of the multiplicative Weyl group action. This orbit space is the image of fundamental invariants and we show that, for the Weyl groups of types A, B, C, D and G, it is a compact basic semi-algebraic set. Furthermore, we give the describing polynomial inequalities through a positive semi-definite Hermite matrix polynomial.Concerning the optimization problem, our rewriting approach transforms the trigonometric optimization problem into a classical polynomial optimization problem on a basic semi-algebraic set. To solve such problems, one can apply Lasserre's hierarchy of moment and sum of squares relaxations. We adapt and implement this hierarchy in the basis of generalized Chebyshev polynomials with a new notion of degree. The optimal value of the original trigonometric optimization problem is then approximated through solutions of semi-definite programs, which are solved numerically.In classical polynomial optimization and other applications, symmetry adaptation has been studied and implemented, but crystallographic symmetries have not been exploited in trigonometric optimization before. With our approach, we provide an alternative to known strategies for trigonometric optimization. Where other techniques utilize sums of Hermitian squares or generalizations of Lasserre's hierarchy to the complex setting, we exploit symmetry first and then apply techniques from classical polynomial optimization. This reduces the size of arising semi-definite relaxations and can also improve the quality of the approximation.As an application, we study the problem of computing lower bounds for the chromatic number of geometric distance graphs. Given a norm and a set of vertices, this problem asks to find the minimal number of colors needed to paint the vertices, so that no two of those of distance 1 between them have the same color. The spectral bound was generalized from finite to infinite graphs to deal with such problems. This bound involves the optimization of a trigonometric polynomial. We focus on norms which are given by polytopes with crystallographic symmetry. Then the problem can be tackled with the techniques developed in this thesis. We give several bounds through analytical and numerical computations., Cette thèse étudie le problème de l'optimisation d'un polynôme trigonométrique avec symétrie cristallographique. L'optimisation des polynômes trigonométriques était l'objet de nombreux travaux récents, mais une théorie permettant d'exploiter les symétries n'a pas été développée ou généralisée. Nous considérons l'action d'un groupe cristallographique, également connu sous le nom de groupe de Weyl, sur les exposants. Cette action admet une définition des polynômes de Chebychev généralisés. Si un polynôme trigonométrique est invariant sous l'action, il peut être écrit comme une somme de polynômes de Chebyshev généralisés dans les invariants fondamentaux.En réécrivant un polynôme trigonométrique invariant comme un polynôme classique, la région d'optimisation est transformée en l'espace d'orbite de l'action du groupe de Weyl multiplicatif. Cet espace d'orbite est l'image des invariants fondamentaux et nous montrons que, pour les groupes de Weyl de types A, B, C, D et G, il s'agit d'un ensemble compact basique semi-algébrique. En plus, nous donnons les inégalités polynomiales descriptives à par un matrix polynomial positif de Hermite.En ce qui concerne l'optimisation, notre approche de réécriture transforme le problème d'optimisation trigonométrique dans un problème d'optimisation polynomiale sur un ensemble basique semi-algébrique. Pour résoudre de tels problèmes, on peut appliquer la hiérarchie des relaxations de moment et de somme de carrés de Lasserre. Nous adaptons et implémentons cette hiérarchie sur la base de polynômes de Chebyshev généralisés avec une nouvelle notion de degré. La valeur optimale du problème d'optimisation trigonométrique original est alors approximée par des solutions de programmes semi-définis, qui sont résolus numériquement.Dans l'optimisation polynomiale classique et d'autres applications, l'adaptation des symétries a été étudiée et implémenté, mais les symétries cristallographiques n'ont pas été exploitées en optimisation trigonométrique auparavant. Avec notre approche, nous fournissons une alternative aux stratégies connues pour l'optimisation trigonométrique. Alors que d'autres techniques utilisent des sommes de carrés Hermitiens ou en généralisant l'hiérarchie de Lasserre au cadre complexe, nous exploitons d'abord la symétrie et appliquons ensuite des techniques d'optimisation polynomiale classique. Cela réduit la taille des relaxations semi-définies qui en découlent et peut également améliorer la qualité de l'approximation.En guise d'application, nous étudions le problème de calcul des bornes inférieurs pour le nombre chromatique des graphes géométriques. Étant donné une norme et un ensemble de sommets, ce problème demande de trouver le nombre minimal de couleurs nécessaires pour peindre les sommets, de sorte que deux de ceux de distance 1 entre eux n'aient pas la même couleur. La borne spectrale a été généralisée des graphes finis aux graphes infinis pour traiter de tels problèmes. Cette limite implique l'optimisation d'un polynôme trigonométrique. Nous nous concentrons sur les normes qui sont données par des polytopes avec une symétrie cristallographique. Le problème peut alors être abordé avec les techniques développées dans cette thèse. Nous donnons plusieurs bornes à travers des calculs analytiques et numériques.

The local community usually has a large heritage; and this a case for the implementation of the ISO 50001 standard. This is due to the fact that the Local Measurement System, recommended by the ISO 50001 standard [1], is not adapted and is not very efficient to identify the Energy Saving Fields Scope (ESFS) in the framework of communities whose asset management is not controlled [2].The objective is to show the limits of the existing system within the framework of the implementation of the Energy Management System (EMS) in the communities and to propose an axis of improvement with an automated Energy Information System, to identify the Certification Scope (CS) for the ISO 50001 standard.In our exploratory approach, we used set and account theories from the mathematical sciences [3] [4] [5] to show the limitations of the Local Measurement System in communities. Then, we created the ESFS (Energy Saving Fields Scope) tool whose main function is to automatically identify the best certification scope for ISO 50001 regardless of the quantity of assets. Tests have shown good results, with numerical simulation data on energy consumption, and further tests are expected with reference data from communities.In the future, the ESFS tool will have to be integrated into an Energy Information System solution., Les collectivités ont, de manière générale, un patrimoine important ; et cela pose des problèmes pour mettre en œuvre la norme ISO 50001. Ce constat s’explique par le fait que le Système de Mesurage Local, préconisé par la norme [1], n’est pas adapté et est inefficace pour identifier le Périmètre de Gisements d’Economie d’Energie dans le contexte des Collectivités dont la gestion du patrimoine n’est pas maitrisée [2].L’objectif ici, est de montrer les limites du système existant dans le cadre d’une mise en œuvre du Système de Management de l’Energie (SMEn) dans les collectivités et de proposer un axe d’amélioration grâce à un Système d’Information Energétique automatisé, pour identifier le Périmètre de Certification (PC) destiné à la norme ISO 50001. Dans notre démarche exploratoire, nous avons utilisé les théories des ensembles et dénombrements des sciences mathématiques [3] [4] [5] pour montrer les limites du Système de Mesurage Local dans les collectivités. Nous avons ensuite réalisé l’outil PGEE (Périmètre de Gisements d’Economie d’Energie) qui a pour fonction principale d’identifier automatiquement le meilleur périmètre de certification dédié à la norme ISO 50001 quelle que soit la quantité de patrimoines. Les tests ont montré de bons résultats, avec des données de simulation numérique concernant la consommation énergétique, et est en attente d’autres tests avec des données de référence des collectivités.Dans une perspective d’évolution, l’outil PGEE devra intégrer le dispositif d’un Système d’Information Energétique.

Using Patterson-Sullivan measures we investigate growth problems for groups acting on a metric space with a strongly contracting element., Comment: 59 pages, 2 figures

This article is a first step in the study of equations in periodic groups. As an application, we study the structure of periodic quotients of hyperbolic groups. We investigate for instance the Hopf and co-Hopf properties, the isomorphism problem, the existence of free splittings, etc. We also consider the automorphism group of such periodic groups., Comment: 123 pages

International audience; Local symmetries are primarily defined in the case of spacetime, but several authors have defined them outside this context, sometimes with the help of groupoids. We show that, in many cases, local symmetries can be defined as global symmetries. We also show that groups can be used, rather than groupoids, to handle local symmetries. Examples are given for graphs and networks, color symmetry and tilings. The definition of local symmetry in physics is also discussed.

The equiangular cubed sphere is a spherical grid, widely used in computational physics. This paper deals with mathematical properties of this grid. We identify the symmetry group,i.e.the group of the orthogonal transformations that leave the cubed sphere invariant. The main result is that it coincides with the symmetry group of a cube. The proposed proof emphasizes metric properties of the cubed sphere. We study the geodesic distance on the grid, which reveals that the shortest geodesic arcs match with the vertices of a cuboctahedron. The results of this paper lay the foundation for future numerical schemes, based on rotational invariance of the cubed sphere.

We define a general notion of centrally $\Gamma$-graded sets and groups and of their graded products, and prove some basic results about the corresponding categories: most importantly, they form braided monoidal categories. Here, $\Gamma$ is an arbitrary (generalized) ring. The case $\Gamma$ = Z/2Z is studied in detail: it is related to Clifford algebras and their discrete Clifford groups (also called Salingaros Vee groups).

Spinors ψj of SU(2) represent group elements, i.e. three-dimensional rotations Rj , because they are shorthands of SU(2) representation matrices Rj obtained by taking their first columns. We explain that making linear combinations of spinors of SU(2) is feasible algebraically but geometrically meaningless. E.g. ψ1 + ψ2 ∈ C 2 would have to correspond to R1 + R2. But in SO(3), R1 + R2 would be a function that transforms r ∈ R 3 into R1(r) + R2(r), while in SU(2) it would correspond to a different function that is up to a multiplication constant equal to S : r → S(r) = R1(r) + R2(r) + 2 [ cos(ϕ1/2) cos(ϕ2/2) − s1•s2 ] r + 2 sin(ϕ1/2) sin(ϕ2/2)[ (s2•r) s1 + (s1•r) s2 ] + 2 sin(ϕ1/2) cos(ϕ2/2)(s1 ∧ r) + 2 cos(ϕ1/2) sin(ϕ2/2)(s2 ∧ r). The multiplication constant must be chosen such that |r| is preserved. Such extensions by linear combinations for spinors from a manifold to an embedding vector space are therefore mindless algebra with a spurious, conceptually impenetrable geometrical counterpart. This should not surprise anybody because the group axioms only define products of group elements, not linear combinations of them. Redefining spinors as vectors of a Hilbert space is therefore a purely formal, would-be scholar generalization.

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

International audience; The Besicovitch pseudodistance defined in [1] for biinfinite sequences is invariant by translations. We generalize the definition to arbitrary locally compact second-countable groups and study how properties of the pseudodistance, including invariance by translations, are determined by those of the sequence of sets of finite positive measure used to define it. In particular, we restate from [2] that if the Besicovitch pseudodistance comes from an exhaustive Følner sequence, then every shift is an isometry. For non-Følner sequences, it is proved that some shifts are not isometries, and the Besicovitch pseudodistance with respect to some subsequences even makes them discontinuous.

This thesis is at the intersection of dynamics, combinatorics and probability theory. My work focuses on a specialization of the notion of amenability: unique ergodicity. This notion refers to those groups whose minimal actions admit a unique invariant measure. I am especially interested in automorphism groups of Fraïssé limits. Those groups have several interesting properties. One of those properties is that they are somewhat "big", meaning not locally compact. The case of locally compact groups is discussed in Chapter 6. Another interesting property of automorphism groups is the understanding we have of some of their actions. This is deeply linked to the fact that we are dealing with automorphism groups. My work essentially relies on the understanding of those actions to construct measures that allow us to understand all the invariant measures of soThis thesis is at the intersection of dynamics, combinatorics and probability theory. My work focuses on a specialization of the notion of amenability: unique ergodicity. This notion refers to those groups whose minimal actions admit a unique invariant measure. I am especially interested in automorphism groups of Fraïssé limits. Those groups have several interesting properties. One of those properties is that they are somewhat "big", meaning not locally compact. The case of locally compact groups is discussed in Chapter 6. Another interesting property of automorphism groups is the understanding we have of some of their actions. This is deeply linked to the fact that we are dealing with automorphism groups. My work essentially relies on the understanding of those actions to construct measures that allow us to understand all the invariant measures of some groups., Cette thèse est à l’intersection de la dynamique, de la combinatoire et de la théorie des probabilités. Mon travail se concentre sur une spécialisation de la notion de moyennabil-ité : l’unique ergodicité. Il s’agit d’une qualification pour décrire les groupes pour lesquelles les actions minimales admettent exactement une mesure invariante. Je m’intéresse partic-ulièrement à des groupes d’automorphismes de limites de Fraïssé qui ont cette propriété. Les groupes d’automorphismes de limites de Fraïssé ont plusieurs propriétés qui les rendent très intéressants. Premièrement, il sont assez "gros", c’est à dire non localement compact. Le chapitre 6 traite le cas des groupes localement compacts. Une autre propriété intéressante est la compréhension que nous avons de la topologie de certains espaces sur lesquels agissent ces groupes, ceci étant directement lié au fait qu’on a à faire à des groupes d’automorphismes. Mon travail consiste principalement à exploiter notre compréhension de ces actions pour con-struire des mesures qui nous permettent de comprendre toutes les mesures invariantes pour certains groupes.

Starting with a separate 2-adic reformulation of the Collatz function that maps parity orbits, we show that the stopping time $k$ of this reformulation can be alternatively derived with the help of some novel ideas as well as hybrid-algorithmic techniques.

We introduce and study a new class of modules and rings we call n/2-perspective, which can either be described in terms of perspective direct summands, associate idempotents, or generalized inverses. When n is small (≤ 2), we recover existing class of modules and rings: (endo)abelian, strongly IC and perspective ones. And 3/2-perspective rings are characterized by all their regular elements being special clean. Standard constructions are also discussed and examples are provided.

In this short note, we study two specific generalized inverses, the (b, c)-inverse of Drazin and the inverse along an element of the present author, in a categorical way. Precisely, we prove that the (b, c)-inverse and the inverse along an element in a semigroup are actually genuine inverse when considered as morphisms in the Schutzenberger category of the semigroup. This category was first defined and studied by Costa and Steinberg, and its definition is based on Green's preoders and relations. In the last section, applications to the Reverse Order Law are given and some new formulas are provided. Examples are also given.

The space of convex projective structures has been well studied with respect to the topological entropy. But, to better understand the geometry of the structure, we study the entropy of the Sinai–Ruelle–Bowen measure and show that it is a continuous function on the space of strictly convex real projective structures.

We classify the (semi-simple parts of the) Lie algebra of the Zariski closure of a discrete subgroup of a split simple real-algebraic Lie group, whose limit sets are minimal and such that the limit set in the space of full flags contains a positive triple of flags (as in Lusztig). We then apply our result to obtain a new proof of Guichard's classification of Zariski closures of Hitchin representations into $PSL_d(\mathbb{R}).$

The mapping class group of an orientable surface with one boundary component, S, is isomorphic to a subgroup of the automorphism group of the fundamental group of S. We call these subgroups algebraic mapping class groups. An algebraic mapping class group acts on a space called ordered Auter space. We apply Poincaré's Polyhedron theorem to this action. We describe a decomposition of ordered Auter space. From these results, we deduce that the algebraic mapping class group of S is a quotient of the fundamental group of a graph of groups with, at most, two vertices and, at most, six edges. Vertex and edge groups of our graph of groups are mapping class groups of orientable surfaces with one, two or three boundary components. A presentation for the mapping class group of S can be obtained by adding, at most, 24 relations to the fundamental group of our graph of groups.

We characterize the Lie groups with finitely many connected components that are $O(u)$-bilipschitz equivalent (almost quasiisometric in the sense that the sublinear function $u$ replaces the additive bounds of quasiisometry) to the real hyperbolic space, or to the complex hyperbolic plane. The characterizations are expressed in terms of deformations of Lie algebras and in terms of pinching of sectional curvature of left-invariant Riemannian metrics in the real case. We also compare sublinear bilipschitz equivalence and coarse equivalence, and prove that every coarse equivalence between the logarithmic coarse structures of geodesic spaces is a $O(\log)$-bilipschitz equivalence. The Lie groups characterized are exactly those whose logarithmic coarse structure is equivalent to that of a real hyperbolic space or the complex hyperbolic plane. Finally we point out that a conjecture made by Tyson about the conformal dimensions of the boundaries of certain hyperbolic buildings holds conditionally to the four exponentials conjecture., Comment: v2->v3:revised, various corrections made according to a referee's work. 50 pages

In this thesis we study groups piecewise acting by isometries on an interval by identifying their abelianization. The case of the Interval Exchange Transformations group, IET, has been done by Arnoux-Fathi and Sah. First, for every subgroup H of the reals, we identify the abelianization of every subgroup IET(H) of the group IET. SImilarly, we identify the abelianization of subgroups of this form of the Interval Exchange Transformations group with flips. The idea is to adapt the group homomorphism signature on finite permutation groups by measuring the set of inversions of an element.Also we prove that the group homomorphism signature on finite permutation groups can be extended to the group consisting of permutations of [0,1[ which are continuous outside a finite number of points. A consequence is the vanishing of an element of the second cohomology group called the Kapoudjian class.Then, we deal with higher dimension. Let d greater than 1 and let REC be the group of all permutations of the unit rectangle in dimension d, which are a translation on a finite number of subrectangles and is the identiy elsewhere. We prove that the natural generalization of restricted rotations (which define a generating subset of IET) define a generating subset of REC. Next we identify the abelianization of REC by extending the work of Arnoux-Fathi and Sah on IET., Le but de cette thèse est d'étudier des groupes agissant par isométries par morceaux sur un intervalle en s'intéressant à leur abélianisé. Le cas du groupe des échanges d'intervalles IET a été traité par Arnoux-Fathi et Sah. Tout d'abord, pour tous sous-groupe H du groupe des réels, on identifie l'abélianisé de tous les sous-groupes IET(H) du groupe IET. On réalise similairement le cas des sous-groupes de cette forme du groupe des échanges d'intervalles avec renversements. L'idée est adapter le morphisme signature sur les groupes de permutations finis en essayant de mesurer l'ensemble des inversions d'un élément. Par ailleurs on démontre aussi que la signature des groupes des permutations finis s'étend au groupe des permutations de de [0,1[ qui sont continus en dehors d'un nombre fini de points. Cela a pour conséquence que la classe de Kapoudjian, un élément du second groupe de cohomologie, s'annule. Ensuite, on se place en dimension d plus grande ou égale que 1 et on considère le groupe REC des permutations du rectangle unité en dimension d, qui bougent un nombre fini de sous-rectangles par translations et qui sont l'identité ailleurs. On démontre que la généralisation naturel des rotations restreintes (qui forment un système de générateur du groupe IET) forme un système générateur de REC. Puis on identifie son abélianisé en généralisant le travail fait pour le groupe IET par Arnoux-Fathi et Sah.

International audience; We study the action of (big) mapping class groups on the first homology of the corresponding surface. We give a precise characterization of the image of the induced homology representation.

42 pages, 2 x 2 figures; We show that several new classes of groups are measure strongly treeable. In particular, finitely generated groups admitting planar Cayley graphs, elementarily free groups, and Isom(H^2) and all its closed subgroups. In higher dimensions, we also prove a dichotomy that the fundamental group of a closed aspherical 3-manifold is either amenable or has strong ergodic dimension 2. Our main technical tool is a method for finding measurable treeings of Borel planar graphs by constructing one-ended spanning subforests in their planar dual. Our techniques for constructing one-ended spanning subforests also give a complete classification of the locally finite p.m.p. graphs which admit Borel a.e. one-ended spanning subforests.

We prove that in the Cayley graph of any braid group modulo its center $B_n/Z(B_n)$, equipped with Garside's generating set, the axes of all pseudo-Anosov braids are strongly contracting. More generally, we consider a Garside group $G$ of finite type with cyclic center. We prove that in the Cayley graph of $G/Z(G)$, equipped with the Garside generators, the axis of any Morse element is strongly contracting. As a consequence, we prove that Morse elements act loxodromically on the additional length graph of $G$., Comment: 25 pages, 9 figures

Developing an idea of M. Gromov, we study the intersection formula for random subsets with density. The \textit{density} of a subset $A$ in a finite set $E$ is defined by $dens A := \log_{|E|}(|A|)$. The aim of this article is to give a precise meaning of Gromov's \textit{intersection formula}: "Random subsets" $A$ and $B$ of a finite set $E$ satisfy $dens (A\cap B) = dens A + dens B -1$. As an application, we exhibit a phase transition phenomenon for random presentations of groups at density $\lambda/2$ for any $0, Comment: version 2, 41 pages

Given a semisimple group over a complete non-Archimedean field, it is well known that techniques from non-Archimedean analytic geometry provide an embedding of the corresponding Bruhat-Tits builidng into the analytic space associated to the group; by composing the embedding with maps to suitable analytic proper spaces, this eventually leads to various compactifications of the building. In the present paper, we give an intrinsic characterization of this embedding., 12 pages, 1 figure. arXiv admin note: text overlap with arXiv:1110.1362

We present a technique to lift some tilings of the discrete hyperbolic plane -- tilings defined by a 1D substitution -- into a zero entropy subshift of finite type (SFT) on non-abelian amenable Baumslag-Solitar groups $BS(1,n)$ for $n\geq2$. For well chosen hyperbolic tilings, this SFT is also aperiodic and minimal. As an application we construct a strongly aperiodic SFT on $BS(1,n)$ with a hierarchical structure, which is an analogue of Robinson's construction on $\mathbb{Z}^2$ or Goodman-Strauss's on $\mathbb{H}_2$.

This thesis offers to study two objects related to singular braid monoids, the desingularization map and the family of Vassiliev invariants, in the respective framework of the singular Artin monoids and the singular virtual braid monoids. In the first chapter, we recall the definitions of these two objects in their original setting, namely the setting of the singular braid monoids which are extensions of the braid groups, before stating the problems which are related to these objects, i.e. deciding whether the desingularization map is one-to-one and whether the Vassilev invariants separate the (classical or singular) braids, as well as the results that answer them. In the second chapter, we give the basics of the theory of Artin groups and the definition of the main families of Artin groups before recalling the definition of the singular extension of an Artin group. We prove a singular version of Van der Lek's theorem on parabolic subgroups of an Artin group for singular Artin monoids of FC type and of affine type. We recall that it is still possible to define in this new framework a desingularization map, Vassiliev invariants and to state the same problems as in the framework of braids, and after giving the results already known about these problems for right angle Artin groups and Artin groups of type I, we prove that the desingularization maps for Artin groups of type A affine are one-to-one and that the Vassiliev invariants respectively of Artin groups of type A affine and of type B separate the elements of these groups. In the third and furth chapter, we recall the definitions of the virtual braid groups, which are other extensions of braid groups, and the singular virtual braid monoids related to them. We show that the singular virtual braid monoids are indeed common extensions of the virtual braid groups and of the singular braid monoids. We then extend to singular virtual braids the combinatorial and topological interpretation (respectively in terms of Gauss diagrams and in terms of stable classes of abstract braids) which was already known for virtual braids. Finally, after defining the desingularization map and the Vassilev invariants for virtual braids, we show that the resolution of the problems related to these objects is equivalent to the resolution of the same problems for a family of Artin groups which appears to be subgroups of the virtual braid groups. We end the thesis with a study on the embeddability of a monoid in its enveloping group, on which several results of the thesis are based on., Cette thèse se propose d’étudier deux objets liés aux monoïdes de tresses singulières, le morphisme de désingularisation et la famille des invariants de Vassiliev, transposés dans les cadres respectifs des monoïdes d’Artin singuliers et des monoïdes de tresses virtuelles singulières. Dans le premier chapitre, on rappelle les définitions de ces deux objets dans leur contexte d'origine, celui des monoïdes de tresses singulières qui sont des extensions des groupes de tresses, puis celles des principales problématiques s'y rapportant, qui sont la déterminations de l'injectivité du morphisme de désingularisation et de la séparabilité des tresses (classiques ou singulières) par les invariants de Vassiliev, ainsi que les résultats qui répondent à ces problématiques. Dans le second chapitre, on redonne les fondamentaux de la théorie des groupes d'Artin et les définitions des principales familles de groupes d'Artin avant de rappeler la définition d'extension singulière associée à un groupe d'Artin. On démontre alors l'analogue singulier du résultat de Van der Lek sur les sous-groupes paraboliques d'un groupe d'Artin pour les monoïdes singuliers d'Artin de type FC et de type affine. On rappelle qu'il est également possible dans ce nouveau contexte de définir un morphisme de désingularisation, des invariants de Vassiliev ainsi que de reformuler les mêmes problématiques que dans le cades des tresses, et après avoir redonné les résultats connus sur ces problématiques pour les groupes d'Artin angle droit et les groupes d'Artin de type I, on démontre l'injectivité du morphisme de désingularisation pour les groupes d'Artin de type A affine ainsi que la séparabilité des éléments des groupes de type A affine et de type B par leurs invariants de Vassiliev respectifs. Dans les troisième et quatrièmes chapitres, on rappelle les définitions des groupes de tresses virtuelles, qui sont d'autres extensions des groupes de tresses, et des monoïdes de tresses virtuelles singulières qui leur sont associés. On démontre que les monoïdes de tresses virtuelles singulières sont bien des extensions communes aux groupes de tresses virtuelles et aux monoïdes de tresses singulières. On étend ensuite aux tresses virtuelles singulières les interprétations combinatoires (en termes de diagrammes de Gauss) et topologiques (en termes de classes stables de tresses abstraites) déjà connues pour les tresses virtuelles. Enfin, après avoir redéfini le morphisme de désingularisation et les invariants de Vassiliev pour les tresses virtuelles, on montre que la résolution des problématiques qui y sont liées est équivalente à la résolution de ces mêmes problématiques pour une famille de groupes d'Artin particuliers qui sont des sous-groupes des groupes de tresses virtuelles. On conclut la thèse par une étude sur le problème de la plongeabilité d'un monoïde dans sa monoïde dans son groupe enveloppant sur laquelle on s'appuie pour démontrer certains résultats de la thèse.

Bass-Serre theory was initiated in 1970 by Jean-Pierre Serre, in [Ser77]. The theory's main motivation was to study the structure of discrete and torsion-free subgroups of SL2(Qp), more precisely Ihara's theorem stating that all torsion-free subgroups of SL2(Qp) are free. Inspired by covering space theory in algebraic topology, J-P Serre explains that showing the freedom of a group by making it act freely on a tree is more natural. So, he deduces a simple and elegant proof of this theorem and allows to generalize several theorems of combinatorial group theory: Nielsen-Schreier, Kurosh and of Nagao theorems and so on. This theory shows more generally that a group acts on a tree without inversion if and only it is isomorphic to a non-trivial amalgam or to an HNN extension.In 2011 B. Deschamps and I. Suarez introduced in [DSA11] a combinatorial theory for profinite groups. They proved an analogue for profinite groups of Serre's theorem on freedom of a group : a profinite group has a dense free subgroup if and only this group acts profreely on a protree. The notion of profree action can be summarized to making the groups of the inverse system of finite groups associated with a profine group act freely on each floor of an inverse system of graphs with certain arithmetic conditions.The purpose of this thesis is to give a contribution of Deschamps-Suarez theory of prographs. Tools and techniques developed by Deschamps and Suarez, placed in a general context, allow to show an analog of the Deschamps-Suarez theorem for profinite groups with a dense amalgamated subgroup and a generalization of this result. Finally, these results are illustrated on well-known Galois situations., La théorie de Bass-Serre a été développée en 1970 par Jean-Pierre Serre, dans [Ser77]. Elle a pour motivation principale d'étudier la structure des sous-groupes discrets et sans torsion de SL2(Qp), plus précisément le théorème d'Ihara suivant lequel ces sous-groupes sont libres. S'inspirant de la topologie notamment de la théorie des revêtements, il devient alors plus naturel, explique J-P Serre de montrer la liberté d'un groupe en le faisant agir librement sur un arbre. Il en déduit ainsi une preuve simple et élégante de ce théorème jugé mystérieux et permet de généraliser plusieurs résultats de théorie combinatoire des groupes : les théorèmes de Nielsen-Schreier, de Kurosh et de Nagao entre autres. Cette théorie montre plus généralement qu'un groupe agit sur un arbre sans inversion si et seulement il est isomorphe à un amalgame non trivial ou à une extension HNN.En 2011 B. Deschamps et I. Suarez ont introduit dans [DSA11] une théorie combinatoire pour les groupes profinis et ont démontré un analogue pour les groupes profinis du théorème de Serre sur liberté d'un groupe : un groupe profini possède un sous-groupe libre dense si et seulement il agit prolibrement sur un poarbre. La notion d'action prolibre se résume moralement à faire agir librement les groupes du système projectif de groupes finis associés à un groupe profini sur chaque étage d’un système projectif de graphes avec certaines conditions arithmétiques.L'objet de cette thèse est de donner une contribution à cette théorie des prographes. Les outils et techniques développés par Deschamps et Suarez étant placé dans un cadre assez général nous permettent alors de montrer un analogue du théorème de DS pour les groupes profinis possédant un sous-groupe amalgamé dense ainsi qu'une généralisation de ce résultat. Enfin nous illustrons ces résultats sur des situations galoisiennes bien connues.

We present a number of examples to illustrate the use of small quotient dessins as substitutes for their often much larger and more complicated Galois (minimal regular) covers. In doing so we employ several useful group-theoretic techniques, such as the Frobenius character formula for counting triples in a finite group, pointing out some common traps and misconceptions associated with them. Although our examples are all chosen from Hurwitz curves and groups, they are relevant to dessins of any type., 32 pages, 16 figures

7 pages; Mirzakhani wrote two papers on counting curves of given type on a surface: one for simple curves, and one for arbitrary ones. We give a complete argument deriving Mirzakhani's result for general curves from the one about simple ones. We then sketch an argument to give a new proof of both results -- full details and other related matters will appear in a book we intend to write.

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