Tesis sobre el tema "Théorie géométrique et ergodique de groupes"
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Long, Yusen. "Diverse aspects of hyperbolic geometry and group dynamics". Electronic Thesis or Diss., université Paris-Saclay, 2024. http://www.theses.fr/2024UPASM016.
Texto completoThis thesis explores diverse topics related to hyperbolic geometry and group dynamics, aiming to investigate the interplay between geometry and group theory. It covers a wide range of mathematical disciplines, such as convex geometry, stochastic analysis, ergodic and geometric group theory, and low-dimensional topology, etc. As research outcomes, the hyperbolic geometry of infinite-dimensional convex bodies is thoroughly examined, and attempts are made to develop integral geometry in infinite dimensions from a perspective of stochastic analysis. The study of big mapping class groups, a current focus in low-dimensional topology and geometric group theory, is undertaken with a complete determination of their fixed-point on compacta property. The thesis also clarifies certain folklore theorems regarding the Gromov hyperbolic spaces and the dynamics of amenable groups on them. Last but not the least, the thesis studies the connectivity of the Gromov boundary of fine curve graphs, a combinatorial tool employed in the study of the homeomorphism groups of surfaces of finite type
Vidotto, Pierre. "Géométrie ergodique et fonctions de comptage en mesure infinie". Nantes, 2016. https://archive.bu.univ-nantes.fr/pollux/show/show?id=464a8384-3383-4967-897b-1160f1741b9a.
Texto completoWe study here some dynamical properties of manifolds M = X/ Γ, endowed with a pinched negative sectional curvature, where X is a Hadamard manifold and Γ = π 1(M) acts by isometries on X. More precisely, we consider divergent Schottky groups Γ whose Bowen-Margulis measure mΓ is infinite on the unit tangent bundle T1X/ Γ. We first define a coding of the action of Γ on the boundary of X, which will be useful to build a symbolic space associated with the geodesic flow. Then we precise the rate of mixing of the geodesic flow (gt)tϵR on T1X/ Γ In a second part, we study the number of closed geodesics on M with length ≤ R. Finally, we give an asymptotic for the orbital counting function # { y ϵ Γ | d(o, y o) ≤ R} when R goes to infinity
Carderi, Alessandro. "Théorie ergodique des actions de groupes et algèbres de von Neumann". Thesis, Lyon, École normale supérieure, 2015. http://www.theses.fr/2015ENSL0995/document.
Texto completoThis dissertation is about measured group theory, sofic entropy and operator algebras. More precisely, we will study actions of groups on probability spaces, some fundamental properties of their sofic entropy (for countable groups), their full groups (for Polish groups) and the amenable subalgebras of von Neumann algebras associated with hyperbolic groups and lattices of Lie groups. This dissertation is composed of three parts.The first part is devoted to the study of sofic entropy of profinite actions. Sofic entropy is an invariant for actions of sofic groups defined by L. Bowen that generalize Kolmogorov's entropy. The definition of sofic entropy makes use of a fixed sofic approximation of the group. We will show that the sofic entropy of profinite actions does depend on the chosen sofic approximation for free groups and some lattices of Lie groups. The second part is based on a joint work with François Le Maître. The content of this part is based on a prepublication in which we generalize the notion of full group to probability measure preserving actions of Polish groups, and in particular, of locally compact groups. We define a Polish topology on these full groups and we study their basic topological properties, such as the topological rank and the density of aperiodic elements. The third part is based on a joint work with Rémi Boutonnet. The content of this part is based on two prepublications in which we try to understand when the von Neumann algebra of a maximal amenable subgroup of a countable group is itself maximal amenable. We solve the question for hyperbolic and relatively hyperbolic groups using techniques due to Popa. With different techniques, we will then present a dynamical criterion which allow us to answer the question for some amenable subgroups of lattices of Lie groups of higher rank
Pinochet, Lobos Antoine. "Théorèmes ergodiques, actions de groupes et représentations unitaires". Thesis, Aix-Marseille, 2019. http://www.theses.fr/2019AIXM0228.
Texto completoIn this thesis, we first study the notion of discrepance, which measures the rate of convergence of ergodic means. We prove estimations for the discrepancy of actions on the sphere, the torus and the Bernoulli shift, as well as for actions of locally compact groups. Moreover, we prove an inequality that allows us to locate these discrepancies in the larger framework of the Monte-Carlo method. We consider the action of the free group on the boundary of its Cayley tree. We prove a convergence theorem of some means associated with this action, that only preserves the class of the natural measures on this boundary. We recover the previously known result that the unitary representation associated to it is irreducible. We then investigate the Howe-Moore property. Groups that satisfy it have the property that whenever they act ergodically on some probability space, then the action is mixing ; unfortunately, this property is not stable by direct products. We formulate a generalization of the Howe-Moore property, relying on an axiomatization of the Mautner phenomenon, that allows us to treat the case of products. Finally, we prove that every lattice inherits the radial rapid decay property, and give an explicit example of a discrete group, endowed with a natural length function which is quasi-isometric to a word-length, that has RRD but doesn't have RD
Lim, Seonhee. "Comptage de réseaux et rigidité entropique pour les actions de groupes sur des arbres et des immeubles". Paris 11, 2006. http://www.theses.fr/2006PA112051.
Texto completoTalbi, Malik. "Inégalité de Haagerup et géométrie des groupes". Lyon 1, 2001. http://www.theses.fr/2001LYO10160.
Texto completoKondah, Abdelaziz. "Les Endomorphismes dilatants de l'intervalle et leurs perturbations aléatoires". Dijon, 1991. http://www.theses.fr/1991DIJOS036.
Texto completoFrancini, Camille. "Caractères de groupes algébriques sur Q et mesures invariantes sur les solénoïdes". Thesis, Rennes 1, 2020. http://www.theses.fr/2020REN1S078.
Texto completoThis thesis is divided in two parts in which the invariant probability measures on solenoids play a major role. The solenoids (that is a compact finite dimensional connected abelian group) are a natural generalization of the usual torus. In the first part, we will study the action of groups on a solenoid by affine transformation; we obtain a necessary and sufficient condition for the action of such a group to have the spectral gap property when the solenoid is provided with the Haar measure. In the second part we will study the trace and characters of algebraic groups over the field of rational numbers. The trace of a countable group are function of positive type on the group which are invariant under conjugation. The characters (that are the indecomposable traces in a certain way) are generalization of the usual characters of finite dimensional representations and intervene in the theory of operator algebra and in the study of invariant random subgroups. We begin with the classification of this characters in the case of unipotent groups. Then we extend this classification to general algebraic groups, using the study of the unipotent case et the establishment of the invariant measure on adelic solenoids
Batakidis, Panagiotis. "Déformation par quantification et théorie de Lie". Paris 7, 2009. http://www.theses.fr/2009PA077149.
Texto completoThe first two chapters review the necessary notions concerning nilpotent Lie algebras, and invariant differential operators. An overview of the existing results of Fujiwara, Corwin-Greenleaf & al. Related to the Duflo conjecture is given in details. The Kontsevich formulation of Deformation Quantization is also reviewed. The generalization for coisotropic submanifolds due to Cattaneo-Felder is also covered and the notion of the reduction algebra is defined. In the third chapter considering a nonzero character of a fixed Lie subalgebra of a general Lie algebra, we prove a Theorem stating that the reduction algebra of the affine space is isomorphic to a deformation of the algebra of invariant differential operators. An explicit formula of this isomorphism is given. Other reduction algebras are examined, and the relations between them and their specializations is described in details. In the forth chapter we calculate characters of the reduction algebra and its specialization thanks to triquantization diagrams and we give an explicit formula for this character. Using double induction, we prove that this character equals the character constructed with the Penney eigendistribution Finally we compute in full detail all the formulas introduced in the text for a 5-dimensional nilpotent Lie algebra. It is shown that the character formula obtained gives an isomorphism between reduction algebras containing the exponential of a differential operator of degree 3 with rational coefficients
Gillibert, Luc. "Aspect géométrique des groupes et des images : les G-graphes et la compression par hypergraphe". Caen, 2006. http://www.theses.fr/2006CAEN2066.
Texto completoThere are two main subject in this thesis : the G-graphs, or the geometrical aspect of the groups, and HLC, or the geometrical aspect of the images applied to the compression. The G-graphs are introduced by Alain Bretto and Alain Faisant in 2003 for studying the group isomorphism problem. But many others applications are possible. We first study the construction of the G-graphs and how groups informations can be visualised on the graph. We gives an algorithm for constructing G-graphs and some theorems for solving the G-graph recognition problem and for the characterisation of bipartite G-graphs. We presents an automatic tool for the recognition of G-graphs and we construct a list of common graphs being G-graphs (Heawood's, Möbius-Kantor's and Dyck's graphs, etc. ). We also work on the classification of symmetric graphs. With G-graphs it is possible to extend the Foster Census, the current reference for cubic symmetric graphs, from the order 768 to the order 1322. We establish some lists of cubic and guintic, symmetric and semisymmetric graphs. Finally we introduce a geometrical representation of the pictures based on rectangle hypergraph. This representation leads ton a lossless compression scheme very efficient on synthetic pictures and named HLC. We show that HLC can be combined with a generic data compression algorithm : PPMd. The choice of PPMd is motivated by an experimental study. We give some experimental results showing the efficiency of HLC+PPMd and we generalise HLC for near-lossless compression and 3D pictures
André, Simon. "Groupes hyperboliques et logique du premier ordre". Thesis, Rennes 1, 2019. http://www.theses.fr/2019REN1S030/document.
Texto completoTwo groups are said to be elementarily equivalent if they satisfy the same first-order sentences in the language of groups, that is the same mathematical statements whose variables are only interpreted as elements of a group. Around 1945, Tarski asked the following question : are non-abelian free groups elementarily equivalent? An affirmative answer to this famous Tarski's problem was given in 2006 by Sela and independently by Kharlampovich and Myasnikov, as the culmination of two voluminous series of papers. Then, Sela gave a classification of all finitely generated groups that are elementarily equivalent to a given torsion-free hyperbolic group. The results contained in the present thesis fall into this context and deal with first-order theories of hyperbolic groups with torsion. In the first chapter, we prove that any finitely generated group that is elementarily equivalent to a hyperbolic group is itself a hyperbolic group. Then, we prove that virtually free groups are almost homogeneous, meaning that elements are almost determined up to automorphism by their type, i.e. the first-order formulas they satisfy. In the last chapter, we give a complete classification of finitely generated virtually free groups up to elementary equivalence with two quantifiers
Pouchin, Guillaume. "Algèbres de Higgs de courbes et cristaux de lacets". Paris 6, 2010. http://www.theses.fr/2010PA066588.
Texto completoBeeker, Benjamin. "Problèmes géométriques et algorithmiques dans des graphes de groupes". Caen, 2011. http://www.theses.fr/2011CAEN2043.
Texto completoThis thesis in geometric group theory gives geometric and algorithmic results on the class of generalized Baumslag-Solitar groups of variable rank (vGBS groups). A vGBS group is one that admits a splitting in a graph of groups where all vertex and edge groups are finitely generated free abelian. We first give a description of the abelian JSJ splittings of vGBS groups. We then describe their abelian compatibility JSJ splittings. We show that, in the class of vGBS groups, the “usual” JSJ splitting is algorithmically constructible, while the compatibility JSJ splitting is not. Finaly we study the multiple conjugacy problem. We show that, although the general problem is undecidable, it is solvable under certain restrictions
Militon, Emmanuel. "Fragmentation et propriétés algébriques des groupes d'homéomorphismes". Phd thesis, Université Paris Sud - Paris XI, 2012. http://tel.archives-ouvertes.fr/tel-00752638.
Texto completoMartin, Alexandre. "Topologie et géométrie des complexes de groupes à courbure négative ou nulle". Phd thesis, Université de Strasbourg, 2013. http://tel.archives-ouvertes.fr/tel-00821442.
Texto completoDe, Loynes Basile. "Graphes et marches aléatoires". Phd thesis, Université Rennes 1, 2012. http://tel.archives-ouvertes.fr/tel-00726483.
Texto completoHorbez, Camille. "Marches aléatoires sur Out(Fn) et sous-groupes d'automorphismes de produits libres". Thesis, Rennes 1, 2014. http://www.theses.fr/2014REN1S114/document.
Texto completoLet G be a countable group that splits as a free product of the form G=G_1*...*G_k*F, where F is a finitely generated free group, and the groups G_i are freely indecomposable and not isomorphic to Z. We show that Out(G) satisfies the Tits alternative, as soon as all the groups G_i and Out(G_i) do. Similar techniques also yield another alternative for subgroups H of Out(F_N), due to Handel and Mosher when H is finitely generated, namely: either H virtually fixes the conjugacy class of some proper free factor of F_N, or H contains a fully irreducible automorphism. Our methods are geometric, and require understanding the dynamics of the action of some subgroups of Out(G) on Gromov hyperbolic spaces. In particular, we determine the closure of the outer space of G relative to the G_i's, as well as the Gromov boundary of the (hyperbolic) complex of relative cyclic splittings of G. We also study random walks on Out(F_N). Given a probability measure mu on Out(F_N) (satisfying some conditions), we prove that almost every sample path of the random walk on (Out(F_N),mu) converges to a point of the Gromov boundary of the free factor complex of F_N, which we identify with the Poisson boundary of (Out(F_N),mu). We also describe the horoboundary of outer space, and give applications to growth of conjugacy classes of F_N under random products of outer automorphisms
Lin, Jyun-Ao. "Algèbres de Hall sphériques de courbes projectives pondérées et algèbres de battage". Sorbonne Paris Cité, 2015. http://www.theses.fr/2015USPCC232.
Texto completoThis thesis has two independent chapters. In the first one, we study the properties of the spherical Hall algebras of coherent sheaves with parabolic structures on a smooth projective curve of arbitrary genus g. We provide a shuffle presentation of this algebra in a combinatorial way and show the existence of some kind of universal spherical Hall algebra of genus g. We alsc prove that the algebra contains ail the characteristic function on the Harder-Narasimhan strata. The second chapter deals with the quantum loop algebra of finite type. We introduce on this algebra a completion, provide a construction of Kashiwara's operators and a new involution. This result is used to construct a conjectured canonical basis on aIl the quantum loop algebras
Matte, Bon Nicolás. "Propriété de Liouville, entropie, et moyennabilité des groupes dénombrables". Thesis, Université Paris-Saclay (ComUE), 2016. http://www.theses.fr/2016SACLS087.
Texto completoThis thesis deals with the Liouville property and amenability of topological full groups of Cantor systems, groups of interval exchanges, and groups acting on rooted trees. In Chapter 2, we provide the first examples of finitely generated, infinite simple groups that have trivial Poisson-Furstenberg boundary for simple random walks (the Liouville property). These arise as the derived subgroup of the topological full groups of a family of minimal subshifts. We show that if the complexity of a (non necessarily minimal) subshift grows strictly subquadratically, every symmetric and finitely supported probability measure on the topological full group has vanishing asymptotic entropy. In Chapter 3, we exhibit a family of topological full groups of minimal subshifts that contain Grigorchuk groups G_ω as subgroups. This shows that the topological full group of a minimal subshift can have subgroups of intermediate growth, answering a question of Grigorchuk. In Chapter 4 (based on a joint work with K. Juschenko, N. Monod, M. de la Salle), we study various features of extensively amenable group actions, a notion which is a tool to prove amenability of groups. As an application, we prove amenability of groups of interval exchanges whose angular components have rational rank at most 2. We also obtain a "Kesten-like" characterisation of extensive amenability in terms of the inverted orbit and use it give a short, probabilistic proof of the fact that recurrent actions are extensively amenable. Finally we study the Liouville property for groups of interval exchanges, and show that there are groups of interval exchanges that admit no finitely supported measure with trivial boundary. In Chapter 5 (based on a joint work with G. Amir, O. Angel, B. Virág), we establish the Liouville property for all groups acting on rooted trees by bounded automorphisms. This includes in particular groups generated by bounded automata. This strengthens results by various authors about amenability of these groups, some of which are based on proving the Liouville property in some special cases
Blossier, Thomas. "Ensembles minimaux localement modulaires : groupes d'automorphismes d'ensembles triviaux et sous-groupes infiniment définissables du groupe additif d'un corps séparablement clos". Paris 7, 2001. http://www.theses.fr/2001PA077172.
Texto completoHafassa, Boutheina. "Deux problèmes de contrôle géométrique : holonomie horizontale et solveur d'esquisse". Thesis, Université Paris-Saclay (ComUE), 2016. http://www.theses.fr/2016SACLS017/document.
Texto completoWe study two problems arising from geometric control theory. The Problem I consists of extending the concept of horizontal holonomy group for affine manifolds. More precisely, we consider a smooth connected finite-dimensional manifold M, an affine connection ∇ with holonomy group H∇ and ∆ a smooth completely non integrable distribution. We define the ∆-horizontal holonomy group H∆∇ as the subgroup of H∇ obtained by ∇-parallel transporting frames only along loops tangent to ∆. We first set elementary properties of H∆∇ and show how to study it using the rolling formalism. In particular, it is shown that H∆∇ is a Lie group. Moreover, we study an explicit example where M is a free step-two homogeneous Carnot group with m≥2 generators, and ∇ is the Levi-Civita connection associated to a Riemannian metric on M, and show in this particular case that H∆∇ is compact and strictly included in H∇ as soon as m≥3. The Problem II is studying the modeling of the problem of solver sketch. This problem is one of the steps of a CAD/CAM software. Our goal is to achieve a well founded mathematical modeling and systematic the problem of solver sketch. The next step is to understand the convergence of the algorithm, to improve the results and to expand the functionality. The main idea of the algorithm is to replace first the points of the space of spheres by displacements (elements of the group) and then use a Newton's method on Lie groups obtained. In this thesis, we classified the possible displacements of the groups using the theory of Lie groups. In particular, we distinguished three sets, each set containing an object type: the first one is the set of points, denoted Points, the second is the set of lines, denoted Lines, and the third is the set of circles and lines, we note that ∧. For each type of object, we investigated all the possible movements of groups, depending on the desired properties. Finally, we propose to use the following displacement of groups for the displacement of points, the group of translations, which acts transitively on Lines ; for the lines, the group of translations and rotations, which is 3-dimensional and acts transitively (globally but not locally) on Lines ; on lines and circles, the group of anti-translations, rotations and dilations which has dimension 4 and acts transitively (globally but not locally) on ∧
Alvarez, Sébastien. "Mesures de Gibbs et mesures harmoniques pour les feuilletages aux feuilles courbées négativement". Phd thesis, Université de Bourgogne, 2013. http://tel.archives-ouvertes.fr/tel-00958080.
Texto completoSage, Marc. "Combinatoire algébrique et géométrique des nombres de Hurwitz". Phd thesis, Université Paris-Est, 2012. http://tel.archives-ouvertes.fr/tel-00804228.
Texto completoJacoboni, Lison. "Propriétés métriques et probabilistes des groupes métabéliens". Thesis, Université Paris-Saclay (ComUE), 2017. http://www.theses.fr/2017SACLS417/document.
Texto completoIn the fist part, we study the return probability of finitely generated metabelian groups. We give a characterization of such groups with large return probability in purely algebraic terms, namely the Krull dimension of the group. To do so, we establish, for metabelian groups, a variation of a famous embedding theorem of Kaloujinine and Krasner that respects this dimension. Along the way, we obtain lower and upper bounds on the return probability of metabelian groups according to their dimension.The second part of this thesis deals with isoperimetric profiles of locally compact compactly generated groups, that we use to characterize the existence of sequences of Følner couples. We generalize at a compact scale previous results of Tessera, in particular that they increase when going to a quotient group, so as to state in more generality a result from the first part, namely that the existence of Følner couples goes to a quotient group. We also prove that it goes to a closed subgroup. This allows to obtains a more self-contained proof of the main result of the first part of this thesis.The third part is a joint work with Kropholler in which we study the structure of soluble groups of infinite torsion-free rank with no ZwrZ. As a corollary, we obtain that a finitely generated soluble group with Krull dimension has finite torsion-free rank if and only if it has no ZwrZ
Bièche, Camille. "Structures de Cauchy-Riemann analytiques et G-stuctures holomorphes associées". Aix-Marseille 1, 2005. http://www.theses.fr/2005AIX11038.
Texto completoMaucourant, François. "Approximation diophantienne, dynamique des chambres de Weyl et répartition d'orbites de réseaux". Phd thesis, Université des Sciences et Technologie de Lille - Lille I, 2002. http://tel.archives-ouvertes.fr/tel-00158036.
Texto completoLa deuxième partie s'intéresse au problème des cibles réctricissantes sur une variété hyperbolique.
Dans la troisième partie, on démontre des résultats de répartition des orbites de l'action de réseaux de groupes de Lie sur certains espaces homogènes, dans la veine de résultats antérieurs de Ledrappier.
Vonseel, Audrey. "Hyperbolicité et bouts des graphes de Schreier". Thesis, Strasbourg, 2017. http://www.theses.fr/2017STRAD025/document.
Texto completoThis thesis is devoted to the study of the topology at infinity of spaces generalizing Schreier graphs. More precisely, we consider the quotient X/H of a geodesic proper hyperbolic metric space X by a quasiconvex-cocompact group H of isometries of X. We show that this quotient is a hyperbolic space. The main result of the thesis indicates that the number of ends of the quotient space X/H is determined by equivalence classes on a sphere of computable radius. In the context of group theory, we show that one can construct explicitly groups and subgroups for which there are no algorithm to determine the number of relative ends. If the subgroup is quasiconvex, we give an algorithm to compute the number of relative ends
Ye, Kaidi. "Automorphismes géométriques des groupes libres : croissance polynomiale et algorithmes". Thesis, Aix-Marseille, 2016. http://www.theses.fr/2016AIXM4713/document.
Texto completoAn automorphism $phi$ of a free group $F_n$ of finite rank $n geq 2$ is said to be geometric it is induced by a homeomorphism on a surface.In this thesis we concern ourselves with answering the question:Which precisely are the outer automorphisms of $F_n$ that are geometric?to which we give an algorithmical decision for the case of polynomially growing outer automorphisms, up to raising to certain positive power.In order to realize this algorithm, we establish the technique of quotient and blow-up automorphisms of graph-of-groups, which when apply for the special case of partial Dehn twist enables us to develop a criterion to decide whether the induced outer automorphism is an actual Dehn twist.Applying the criterion repeatedly on the special topological representative deriving from relative train track map, we are now able to either “unfold” this iterated relative Dehn twist representative level by level until eventually obtain an ordinary Dehn twist representative or show that $hat{phi}$ has at least quadratic growth hence is not geometric.As a side result, we also proved that every linearly growing automorphism of free group has a positive power which is a Dehn twist automorphism. This is a fact that has been taken for granted by many experts, although has no formal proof to be found in the literature.In the case of Dehn twist automorphisms, we then use the known algorithm to make the given Dehn twist representative efficient and apply the Whitehead algorithm as well as the classical theorems by Nielsen, Baers, Zieschangs and others to construct its geometric model or to show that it is not geometric
Hils, Martin. "Fusion libre et autres constructions génériques". Phd thesis, Université Paris-Diderot - Paris VII, 2006. http://tel.archives-ouvertes.fr/tel-00274128.
Texto completola fusion libre sur une fusion fortement minimale est effectuée. Puis, des variations sur le thème de la fusion sont étudiées (courbe générique et structures bicolores). À titre d'exemple, il suit des résultats que l'on peut donner un sens à la notion d'une courbe générique dans un corps pseudofini. Enfin, l'axiomatisabilité de l'automorphisme générique est démontrée dans certains contextes issus d'une amalgamation à la Hrushovski dont la fusion libre et les théories des différents corps bicolores de Poizat (noir, rouge et vert).
Lazrag, Ayadi. "Théorie de contrôle et systèmes dynamiques". Thesis, Nice, 2014. http://www.theses.fr/2014NICE4060/document.
Texto completoThis thesis is devided into three parts. In the first part we begin by describing some well known results in geometric control theory such as the Chow Rashevsky Theorem, the Kalman rank condition, the End-Point Mapping and the linear test. Moreover, we define and study briefly local controllability around a reference control at first and second order. In the second part we provide an elementary proof of the Franks lemma for geodesic flows using basic tools of geometric control theory. In the last part, given a compact Riemannian manifold, we prove a uniform Franks' lemma at second order for geodesic flows and apply the result in persistence theory. In this part we introduce with more details notions of local controllability at first and second order. In fact, we provide a second order controllability result whose proof is long and technical
Marquis, Ludovic. "Les pavages en géométrie projective de dimension 2 et 3". Phd thesis, Université Paris Sud - Paris XI, 2009. http://tel.archives-ouvertes.fr/tel-00428902.
Texto completoKaya, Oğuzhan. "Espaces symétriques compacts, quantification et représentations". Thesis, Lille 1, 2015. http://www.theses.fr/2015LIL10092.
Texto completoLet U be a compact Lie group and K a closed subgroup such that the quotient space U/K is a compact symmetric space. We apply geometric quantization to the cotangent bundle of U/K, for which we have two natural choices for a polarization: the vertical polarization and the holomorphic polarization. Geometric quantization then gives us two Hilbert spaces, one a space of functions on U/K and the other a space of holomorphic functions on the cotangent bundle which can be identifies with the complexification U^C/K^C. We obtain the BKS pairing between these two spaces, which gives us (in theory) the BKS transform between these two spaces. In order to study whether this BKS transform is unitary, we use in particular the theory of unitary representations of compact Lie groups, which allows us to reduce this question to a question that involves only spherical functions. Uniqueness of spherical functions for a given representation then simplifies the problem enormously. Our method can be applied to compact Lie groups by considering a compact Lie group K as a compact symmetric space: it suffices to take U=K x K and K =diag(K). By using Kirillov's character formula our method allows us to give yet another proof that the BKS transform for compact Lie groups is unitary (our proof is a variation of the proof given by Huebschmann). Our method also allows us to show that the BKS transform is not unitary for an arbitrary compact symmetric space. An explicit counter example is the 5-sphere S^5 seen as symmetric space by taking S^5=SO(6)/SO(5). On the other hand, introducing the parameter hbar as suggested by physics, we show that the BKS transform is asymptotically unitary for all compact symmetric spaces in the limit hbar --> 0
Carette, Mathieu. "The automorphism group of accessible groups and the rank of Coxeter groups". Doctoral thesis, Universite Libre de Bruxelles, 2009. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/210261.
Texto completoVia la théorie de Bass-Serre, un groupe agissant sur un arbre est doté d'une structure algébrique particulière, généralisant produits amalgamés et extensions HNN. Le groupe est en fait déterminé par certaines données combinatoires découlant de cette action, appelées graphes de groupes.
Un cas particulier de cette situation est celle d'un produit libre. Une présentation du groupe d'automorphisme d'un produit libre d'un nombre fini de groupes librement indécomposables en termes de présentation des facteurs et de leurs groupes d'automorphismes a été donnée par Fouxe-Rabinovich. Il découle de son travail que si les facteurs et leurs groupes d'automorphismes sont de présentation finie, alors le groupe d'automorphisme du produit libre est de présentation finie. Une première partie de cette thèse donne une nouvelle preuve de ce résultat, se basant sur le langage des actions de groupes sur les arbres.
Un groupe accessible est un groupe de type fini déterminé par un graphe de groupe fini dont les groupes d'arêtes sont finis et les groupes de sommets ont au plus un bout, c'est-à-dire qu'ils ne se décomposent pas en produit amalgamé ni en extension HNN sur un groupe fini. L'étude du groupe d'automorphisme d'un groupe accessible est ramenée à l'étude de groupes d'automorphismes de produits libres, de groupes de twists de Dehn et de groupes d'automorphismes relatifs des groupes de sommets. En particulier, on déduit un critère naturel pour que le groupe d'automorphismes d'un groupe accessible soit de présentation finie, et on donne une caractérisation des groupes accessibles dont le groupe d'automorphisme externe est fini. Appliqués aux groupes hyperboliques de Gromov, ces résultats permettent d'affirmer que le groupe d'automorphismes d'un groupe hyperbolique est de présentation finie, et donnent une caractérisation précise des groupes hyperboliques dont le groupe d'automorphisme externe est fini.
Enfin, on étudie le rang des groupes de Coxeter, c'est-à-dire le cardinal minimal d'un ensemble générateur pour un groupe de Coxeter donné. Plus précisément, on montre que si les composantes de la matrice de Coxeter déterminant un groupe de Coxeter sont suffisamment grandes, alors l'ensemble générateur standard est de cardinal minimal parmi tous les ensembles générateurs.
Doctorat en Sciences
info:eu-repo/semantics/nonPublished
Hoarau, Emma. "Mise en évidence de la brisure de symétrie des schémas numériques pour l'aérodynamique et développement de schémas préservant ces symétries". Paris 6, 2009. http://www.theses.fr/2009PA066650.
Texto completoGay, Joël. "Representation of Monoids and Lattice Structures in the Combinatorics of Weyl Groups". Thesis, Université Paris-Saclay (ComUE), 2018. http://www.theses.fr/2018SACLS209/document.
Texto completoAlgebraic combinatorics is the research field that uses combinatorial methods and algorithms to study algebraic computation, and applies algebraic tools to combinatorial problems. One of the central topics of algebraic combinatorics is the study of permutations, interpreted in many different ways (as bijections, permutation matrices, words over integers, total orders on integers, vertices of the permutahedron…). This rich diversity of perspectives leads to the following generalizations of the symmetric group. On the geometric side, the symmetric group generated by simple transpositions is the canonical example of finite reflection groups, also called Coxeter groups. On the monoidal side, the simple transpositions become bubble sort operators that generate the 0-Hecke monoid, whose algebra is the specialization at q=0 of Iwahori’s q-deformation of the symmetric group. This thesis deals with two further generalizations of permutations. In the first part of this thesis, we first focus on partial permutations matrices, that is placements of pairwise non attacking rooks on a n by n chessboard, simply called rooks. Rooks generate the rook monoid, a generalization of the symmetric group. In this thesis we introduce and study the 0-Rook monoid, a generalization of the 0-Hecke monoid. Its algebra is a proper degeneracy at q = 0 of the q-deformed rook monoid of Solomon. We study fundamental monoidal properties of the 0-rook monoid (Green orders, lattice property of the R-order, J-triviality) which allow us to describe its representation theory (simple and projective modules, projectivity on the 0-Hecke monoid, restriction and induction along an inclusion map).Rook monoids are actually type A instances of the family of Renner monoids, which are completions of the Weyl groups (crystallographic Coxeter groups) for Zariski’s topology. In the second part of this thesis we extend our type A results to define and give a presentation of 0-Renner monoids in type B and D. This also leads to a presentation of the Renner monoids of type B and D, correcting a misleading presentation that appeared earlier in the litterature. As in type A we study the monoidal properties of the 0-Renner monoids of type B and D : they are still J-trivial but their R-order are not lattices anymore. We study nonetheless their representation theory and the restriction of projective modules over the corresponding 0-Hecke monoids. The third part of this thesis deals with different generalizations of permutations. In a recent series of papers, Châtel, Pilaud and Pons revisit the algebraic combinatorics of permutations (weak order, Malvenuto-Reutenauer Hopf algebra) in terms of the combinatorics of integer posets. This perspective encompasses as well the combinatorics of quotients of the weak order such as binary trees, binary sequences, and more generally the recent permutrees of Pilaud and Pons. We generalize the weak order on the elements of the Weyl groups. This enables us to describe the order on vertices of the permutahedra, generalized associahedra and cubes in the same unified context. These results are based on subtle properties of sums of roots in Weyl groups, and actually fail for non-crystallographic Coxeter groups
Hernández, Hernández Jesús. "Combinatorial rigidity of complexes of curves and multicurves". Thesis, Aix-Marseille, 2016. http://www.theses.fr/2016AIXM4707.
Texto completoSuppose S = Sg,n is an orientable connected surface of finite topological type, with genus g ≥ 3 and n ≥ 0 punctures. In the first two chapters we describe the principal set of a surface, and prove that through iterated rigid expansions we can create an increasing sequence of finite sets whose union in the curve complex of the surface C(S). In the third chapter we introduced Aramayona and Leininger's finite rigid set X(S) and use it to prove that the increasing sequence of the previous two chapters becomes an increasing sequence of finite rigid sets after, at most, the fifth iterated rigid expansion. We use this to prove that given S1 = Sg1,n1 and S2 = Sg2,n2 surfaces such that k(S1) ≥ k(S2) and g1 ≥ 3, any edge-preserving map from C(S1) to C(S2) is induced by a homeomorphism from S1 to S2. This is later used to prove a similar statement using homomorphisms from certain subgroups of Mod*(S1) to Mod*(S2). In the fourth chapter we use the previous results to prove that the only way to obtain an edge-preserving and alternating map from the Hatcher-Thurston graph of S1 = Sg,0, HT(S1), to the Hatcher-Thurston graph of S2 = Sg,n, HT(S2), is using a homeomorphism of S1 and then make n punctures to the surface to obtain S2. As a consequence, any edge-preserving and alternating self-map of HT(S) as well as any automorphism is induced by a homeomorphism. In the fifth chapter we prove that any superinjective map from the nonseparating and outer curve graph of S1, NO(S1), to that of S2, NO(S2), is induced by a homeomorphism assuming the same conditions as in the previous chapters. Finally, in the conclusions we discuss the meaning of these results and possible ways to expand them