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Cannizzo, Jan. "Schreier Graphs and Ergodic Properties of Boundary Actions." Thesis, Université d'Ottawa / University of Ottawa, 2014. http://hdl.handle.net/10393/31444.
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This thesis is broadly concerned with two problems: investigating the ergodic properties of boundary actions, and investigating various properties of Schreier graphs. Our main result concerning the former problem is that, in a variety of situations, the action of an invariant random subgroup of a group G on a boundary of G (e.g. the hyperbolic boundary, or the Poisson boundary) is conservative (there are no wandering sets). This addresses a question asked by Grigorchuk, Kaimanovich, and Nagnibeda and establishes a connection between invariant random subgroups and normal subgroups. We approach the latter problem from a number of directions (in particular, both in the presence and the absence of a probability measure), with an emphasis on what we term Schreier structures (edge-labelings of a given graph which turn it into a Schreier coset graph). One of our main results is that, under mild assumptions, there exists a rich space of invariant Schreier structures over a given unimodular graph structure, in that this space contains uncountably many ergodic measures, many of which we are able to describe explicitly.
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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.
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Cette thèse explore divers sujets liés à la géométrie hyperbolique et à la dynamique de groupes, dans le but d'étudier l'interaction entre la géométrie et la théorie de groupes. Elle couvre un large éventail de disciplines mathématiques, telles que la géométrie convexe, l'analyse stochastique, la théorie ergodiques et géométriques de groupes, et la topologie en basses dimensions, et cætera. Comme résultats de recherche, la géométrie hyperbolique des corps convexes en dimension infinie est examinée en profondeur, et des tentatives sont faites pour développer la géométrie intégrale en dimension infinie d'un point de vue de l'analyse stochastique. L'étude des gros groupes de difféotopies, un sujet d'actualité en topologie en basses dimensions et en théorie géométrique de groupes, est entreprise avec une détermination complète de leur propriété de point fixe sur les compacts. La thèse étudie la connexité du bord de Gromov des graphes de courbes fins, un outil combinatoire utilisé dans l'étude des groupes d'homéomorphismes des surfaces de type fini. Enfin, la thèse clarifie également certains théorèmes folkloriques concernant les espaces hyperboliques au sens de Gromov et la dynamique des groupes moyennables sur ces espacesThis 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
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Benson, Martin. "Topics in geometric group theory." Thesis, University of Nottingham, 2005. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.428957.
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Williams, Benjamin Thomas. "Two topics in geometric group theory." Thesis, University of Southampton, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.323942.
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Isenrich, Claudio Llosa. "Kähler groups and Geometric Group Theory." Thesis, University of Oxford, 2017. https://ora.ox.ac.uk/objects/uuid:4a7ab097-4de5-4b72-8fd6-41ff8861ffae.
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In this thesis we study Kähler groups and their connections to Geometric Group Theory. This work presents substantial progress on three central questions in the field: (1) Which subgroups of direct products of surface groups are Kähler? (2) Which Kähler groups admit a classifying space with finite (n-1)-skeleton but no classifying space with finitely many n-cells? (3) Is it possible to give explicit finite presentations for any of the groups constructed in response to Question 2? Question 1 was raised by Delzant and Gromov. Question 2 is intimately related to Question 1: the non-trivial examples of Kähler subgroups of direct products of surface groups never admit a classifying space with finite skeleton. The only known source of non-trivial examples for Questions 1 and 2 are fundamental groups of fibres of holomorphic maps from a direct product of closed surfaces onto an elliptic curve; the first such construction is due to Dimca, Papadima and Suciu. Question 3 was posed by Suciu in the context of these examples. In this thesis we: provide the first constraints on Kähler subdirect products of surface groups (Theorem 7.3.1); develop new construction methods for Kähler groups from maps onto higher-dimensional complex tori (Section 6.1); apply these methods to obtain irreducible examples of Kähler subgroups of direct products of surface groups which arise from maps onto higher-dimensional tori and use them to show that our conditions in Theorem 7.3.1 are minimal (Theorem A); apply our construction methods to produce irreducible examples of Kähler groups that (i) have a classifying space with finite (n-1)-skeleton but no classifying space with finite n-skeleton and (ii) do not have a subgroup of finite index which embeds in a direct product of surface groups (Theorem 8.3.1); provide a new proof of Biswas, Mj and Pancholi's generalisation of Dimca, Papadima and Suciu's construction to more general maps onto elliptic curves (Theorem 4.3.2) and introduce invariants that distinguish many of the groups obtained from this construction (Theorem 4.6.2); and, construct explicit finite presentations for Dimca, Papadima and Suciu's groups thereby answering Question 3 (Theorem 5.4.4)).
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Ashdown, M. A. J. "Geometric algebra, group theory and theoretical physics." Thesis, University of Cambridge, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.596181.
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This dissertation applies the language of geometric algebra to group theory and theoretical physics. Geometric algebra, which is introduced in Chapter 2, provides a natural extension of the concept of multiplication from real numbers to geometric objects such as line segments and planes. It is based on Clifford algebra and augmented by auxiliary definitions which give it a geometric interpretation. Since geometric algebra provides a natural encoding of the concepts of directed quantities, it has the potential to unify many of the disparate systems of notation that are used in mathematics. In Chapter 3, the properties of multilinear functions are investigated and the theory is developed to make them useful for formulating the representation of groups. It will be found that multilinear functions are more flexible than their tensor or matrix counterparts in traditional linear algebra. Multilinear functions can be classified according to the symmetry class of their arguments and their behaviour under the monogenic or harmonic decomposition. It is found that the previous definitions of monogenic and harmonic functions need some modification if they are to be defined consistently. Polynomial projection is also discussed, a technique that is useful in constructing non-linear functions from linear functions, an operation outside the scope of conventional linear algebra. In Chapter 4, multilinear functions are used to construct the irreducible representations of the three regular classes of classical groups; rotation groups, the special unitary and special linear group, and the symplectic group. In each case it is found that a decomposition must be applied to the multilinear functions in order to find the irreducible representations of the groups. For the representations of some of the groups this entails finding the harmonic or monogenic parts of the functions. The groups can be realised as subgroups of the spin group of some dimension and signature. However, geometric algebra provides such a rich algebraic structure that the representations of the groups can be realised in more than one way. In Chapter 7 a brief review is given of computer software for performing symbolic calculations with geometric algebra. A new software package which performs semi-symbolic manipulation of multivectors in spaces of any dimension and signature is presented.
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Gill, Olivia Jo. "Geometric and homological methods in group theory : constructing small group resolutions." Thesis, London Metropolitan University, 2011. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.573402.
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Given two groups K and H for which we have the free crossed resolutions, B* ɛ K and C* ɛ H respectively. Our aim is to construct a free crossed resolution, A* ɛ G, by way of induction on the degree n, for any semidirect product G = K >
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Joubert, Paul. "Geometric actions of the absolute Galois group." Thesis, Stellenbosch : University of Stellenbosch, 2006. http://hdl.handle.net/10019.1/2508.
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Thesis (MSc (Mathematics))--University of Stellenbosch, 2006.This thesis gives an introduction to some of the ideas originating from A. Grothendieck's 1984 manuscript Esquisse d'un programme. Most of these ideas are related to a new geometric approach to studying the absolute Galois group over the rationals by considering its action on certain geometric objects such as dessins d'enfants (called stick figures in this thesis) and the fundamental groups of certain moduli spaces of curves. I start by defining stick figures and explaining the connection between these innocent combinatorial objects and the absolute Galois group. I then proceed to give some background on moduli spaces. This involves describing how Teichmuller spaces and mapping class groups can be used to address the problem of counting the possible complex structures on a compact surface. In the last chapter I show how this relates to the absolute Galois group by giving an explicit description of the action of the absolute Galois group on the fundamental group of a particularly simple moduli space. I end by showing how this description was used by Y. Ihara to prove that the absolute Galois group is contained in the Grothendieck-Teichmuller group.
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El-Mosalamy, Mohamed Soliman Hassan. "Applications of star complexes in group theory." Thesis, University of Glasgow, 1987. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.293464.
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Fennessey, Eric James. "Some applications of geometric techniques in combinatorial group theory." Thesis, University of Glasgow, 1989. http://theses.gla.ac.uk/6159/.
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Combinatorial group theory abounds with geometrical techniques. In this thesis we apply some of them to three distinct areas. In Chapter 1 we present all of the techniques and background material neccessary to read chapters 2,3,4. We begin by defining complexes with involutary edges and define coverings of these. We then discuss equivalences between complexes and use these in §§1.3 and 1.4 to give a way (the level method) of simplifying complexes and an application of this method (Theorem 1.3). We then discuss star-complexes of complexes. Next we present background material on diagrams and pictures. The final section in the chapter deals with SQ-universality. The.basic discussion of complexes is taken from notes, by Pride, on complexes without involutary edges, and modified by myself to cover complexes with involution. Chapters 2,3, and 4 are presented in the order that the work for them was done. Chapters 2,3, alld 4 are intended (given the material in chapter 1) to be self contained, and (iv) each has a full introduction. In Chapter 2 we use diagrams and pictures to study groups with the following structure. (a) Let r be a graph with vertex set V and edge set E. We assume that no vertex of r is isolated. (b) For each vertex VEV there is a non-trivial group Gv ' (c) For each edge e-{u,v}EE there is a set Se of cyclically reduced elements of Gu*Gv , each of length at least two. We define Ge to be the quotient of Gu*Gv by the normal closure of Se. We let G be the quotient of *Gv by the normal closure of VEV S- USe. For convenience, we write eEE The above is a generalization ofa situation studied by Pride [35], where each Gv was infinite cyclic.' Let e-{u,v} be an edge of r. We will say that Ge has property-Wk if no non-trivial element of Gu*Gv of free product length less than or equal to 2k is in the kernel of the natural epimorphism (v) We will work with one of the following: (I) Each Ge has property-W2 (II) r is triangle-free and each Ge has property-WI' Assuming that (I) or (II) holds we: (i) prove a Freihietssatz for these groups; (ii) give sufficient conditions for the groups to be SQ-universal; (iii) prove a result which allows us to give long exact sequences relating the (co)-homology G to the (co)-homology of the groups The work in Chapter 2 is in some senses the least original. The proofs are extensions of proofs given in [35] and [39] for the case when each Gv is infinite cyclic. However. there are some technical difficulties which we had to overcome. In chapter 3 we use the two ideas of star-complexes and coverings to look at NEC-groups. An NEC (Non-Euclidean Crystallographic) group is a discontinuous group of isometries (some of which may be (vi) orientation reversing) of the Non-Euclidean plane. According to Yilkie [46], a finitely generated NEC-group with compact orbit space has a presentation as follows: Involutary generators: Yij (i,j)EZo Non-involutary generators: 6i (iElf), tk (l~~r) (*) Defining paths: (YijYij+,)mij (iElf, l~j~n(i)-l) where In Hoare, Karrass and Solitar [22] it is shown that a subgroup of finite index in a group with a presentation of the form (*), has itself a presentation of the form (*). In [22] the same authors show that a subgroup of infinite ingex in a group with a presentation of the form (*) is a free product of groups of the following types: (A) Cyclic groups. (vii) (B) Groups with presentations of the form Xl' ... 'Xn involutary. (e) Groups with presentations of the form Xi (iEZ) involutary. We define what we mean by an NEe-complex. (This involves a structural re$triction on the form of the star-complex of the complex.) It is obvious from the definition that this class of complexes is clo$ed under coverings, so that the class of fundamental groups of NEe-complexes is trivially closed under taking subgroups. We then obtain structure theorems for both finite and infinite NEe-complexes. We show that the fundamental group of a finite NEe-complex has a presentation of the form (*) and that the fundamental group of an infinite NEe-complex is a free product of groups of the forms (A). (B) and (e) above. We then use coverings to derive some of the results on normal subgroups of NEe-groups given in [5] and [6]. , (viii) In chapter 4 we use the techniques of coverings and diagrams. to stue,iy the SQ-universau'ty of Coxeter groups. This is a problem due to B.H. Neumann (unpublished). see [40]. A Coxeter pair is a 2-tup1e (r.~) where r is a graph (with vertex set V(r) and edge set E(r» and ~ is a map from E(r) to {2.3.4 •.•• }. We associate with (r.~) the Coxeter group c(r,~) defined by the presentation tr(r,~)-
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Bergeron, Maxime. "On CAT(0) aspects of geometric group theory and some applications to geometric superrigidity." Thesis, McGill University, 2012. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=110571.
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Since their popularization by Gromov in the eighties, CAT(0) metric spaces of bounded curvature as defined by Alexandrov have been the locus of great progress in infinite group theory. Surveying ideas and constructions of geometric group theory, we express a bias towards groups acting on structures of this kind. As such, swiftly acquainting the reader with the theory of CAT(0) spaces, we provide a variety of examples obtained by gluing together families of convex polyhedra along their isometric faces. In this context, Gromov's link condition provides a local-to-global framework for non-positive curvature. Combining this with tools from knot theory, such as the Dehn complex of an alternating knot projection, we demonstrate a result of Wise which states that the fundamental group of an alternating link complement is also the fundamental group of a non-positively curved complex. Using similar ideas, we also mention a construction of Wise relating any finitely generated group to the fundamental groups of some non-positively curved complexes. Besides providing such "explicit" constructions, we make use of tower lifts of combinatorial maps to prove Bridson and Haefliger's abstract result that every subgroup of the fundamental group of a non-positively curved two dimensional polyhedral complexes is the fundamental group of some compact non-positively curved two dimensional polyhedral complex. Then, having well established the inherent structure of CAT(0) spaces, we focus on classifying their isometries, group actions upon them, and how they extend to the visual boundary. The combinatorial approach is especially effective here when we prove Haglund's result that cell-preserving isometries of CAT(0) cube complexes are semi-simple.Finally, using the theory of generalized harmonic maps, we demonstrate the superrigidity result of Monod, Gelander, Karlsson and Margulis for reduced actions with no globally fixed point of irreducible uniform lattices in locally compact, compactly generated topological groups of higher rank on complete CAT(0) spaces.Depuis leur popularisation par Gromov durant les années quatre-vingt, la théorie des espaces métriques à courbure bornée, dits CAT(0), fut à la base de grandes percées dans notre compréhension des groupes infinis. Survolant des constructions de la théorie géométrique des groupes, nous portons donc une attention particulière aux actions sur les espaces CAT(0) et commençcons notre traité par la construction de complexes CAT(0) obtenus en identifiant certaines faces isométriques d'ensembles de polyèdres convexes. Dans ce contexte, le critère du lien de Gromov nous permet de caractériser la courbure nonpositive globale de manière locale. Combinant ces idées à certaines techniques de la théorie des noeuds, nous démontrons un théorème de Wise reliant tout groupe fondamental du complément d'un entrelac alternants à un complexe de courbure nonpositive. Nous relatons aussi une construction similaire de Wise permettant de relier tout groupe présenté de manière finie au groupe fondamental d'un complexe à courbure nonpositive. Outre ces constructions concrètes, nous utilisons les tours de relèvement d'applications combinatoires afin de démontrer un théorème abstrait de Bridson et Haefliger concernant les sous-groupes de groupes fondamentaux de complexes à courbure non-positive. Ayant établi la structure des espaces CAT(0), nous passons en second lieu à la classification de leurs isométries et de leurs extensions à la bordification de ces espaces. L'approche combinatoire est d'une aide particulière lorsque nous prouvons le résultat de Haglund concernant la semi-simplicité d'isométries de complexes cubiques et offre un contraste par rapport à un résultat analogue de Brisdon dans le contexte des complexes polyhédraux. Finalement, en faisant usage de la théorie des applications harmoniques généralisées, nous démontrons le résultat de superrigidité de Monod, Gelander, Karlsson et Margulis pour les actions réduites sans point fixe sur les espaces métriques CAT(0) complets de réseaux uniformes et irréductibles dans des groupes de rang supérieur localement compacts engendrés par un ensemble de générateurs compact.
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Jarrett, Kieran. "Non-singular actions of countable groups." Thesis, University of Bath, 2018. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.761021.
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In this thesis we study actions of countable groups on measure spaces underthe assumption that the dynamics are non-singular, with particular reference topointwise ergodic theorems and their relationship to the critical dimensions ofthe action.
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Bavuma, Yanga. "Some combinatorial aspects in algebraic topology and geometric group theory." Master's thesis, University of Cape Town, 2018. http://hdl.handle.net/11427/29763.
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The present Msc thesis deals with classical topics of topology and it has been written, referring to [C. Kosniowski, Introduction to Algebraic Topology, Cambridge University Press, 1980, Cambridge], which is a well known textbook of algebraic topology. It has been selected a list of main exercises from this reference, whose solutions were not directly available, or subject to differerent methods. In fact combinatorial methods have been preferred and the result is a self-contained dissertation on the theory of the fundamental group and of the coverings. Finally, there are some recent problems in geometric group theory which are related to the presence of finitely presented groups which appear naturally as fundamental groups.
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Davidson, Peter John. "Geometric methods in the study of Pride groups and relative presentations." Thesis, Connect to e-thesis, 2008. http://theses.gla.ac.uk/230/.
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Thesis (Ph.D.) - University of Glasgow, 2008.Ph.D. thesis submitted to the Faculty of Information and Mathematical Sciences, Department of Mathematics, University of Glasgow, 2008. Includes bibliographical references. Print version also available.
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Sathaye, Bakul Sathaye. "Obstructions to Riemannian smoothings of locally CAT(0) manifolds." The Ohio State University, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=osu1531416481628579.
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Atanasov, Risto. "Groups of geometric dimension 2." Diss., Online access via UMI:, 2007.
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Cotton-Barratt, Owen. "Geometric and profinite properties of groups." Thesis, University of Oxford, 2011. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.572875.
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We use profinite Bass-Serre theory (the theory of profinite group actions on profinite trees) to prove that the fundamental groups of finite graphs of free groups which are l-acylindrical and have finitely generated edge groups are conjugacy separable. We apply this theorem to: demonstrate that a generic positive one-relator group is conjugacy separable; produce a variant of the Rips con- struction in which the output group is conjugacy separable; apply this last to exhibit an example of a strong profinite equivalence between two finitely presented groups, one of which is conjugacy separable and the other having unsolvable conjugacy problem. We further use profinite Bass-Serre theory to demonstrate that having one end is an up-weak pro-C property for any extension- closed class C of finite groups. We show by example that it is not a down-weak pro-p property for any prime p. We consider Korenev's definition of pro-p ends for a pro-p group, and show that the number of ends of a finitely generated residually p group cannot be less than the number of pro-p ends of its pro-p completion. We explore possibilities for, but are ultimately unsuc- cessful in giving, a proper analogue of Stallings' theorem for pro-p groups. We ask which other properties might be profinite, and use another variant of the Rips construction to produce examples of patholog- ical groups such that either they are hyperbolic groups which are not residually finite, or neither property (FA) nor property (T) is an up-weak profinite property.
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Genevois, Anthony. "Cubical-like geometry of quasi-median graphs and applications to geometric group theory." Thesis, Aix-Marseille, 2017. http://www.theses.fr/2017AIXM0569/document.
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La classe des graphes quasi-médians est une généralisation des graphes médians, ou de manière équivalente, des complexes cubiques CAT(0). L'objectif de cette thèse est d'introduire ces graphes dans le monde de la théorie géométrique des groupes. Dans un premier temps, nous étendons la notion d'hyperplan définie dans les complexes cubiques CAT(0), et nous montrons que la géométrie d'un graphe quasi-médian se réduit essentiellement à la combinatoire de ses hyperplans. Dans la deuxième partie de notre texte, qui est le cœur de la thèse, nous exploitons la structure particulière des hyperplans pour démontrer des résultats de combinaison. L'idée principale est que si un groupe agit d'une bonne manière sur un graphe quasi-médian de sorte que les stabilisateurs de cliques satisfont une certaine propriété P de courbure négative ou nulle, alors le groupe tout entier doit satisfaire P également. Les propriétés que nous considérons incluent : l'hyperbolicité (éventuellement relative), les compressions lp (équivariantes), la géométrie CAT(0) et la géométrie cubique. Finalement, la troisième et dernière partie de la thèse est consacrée à l'application des critères généraux démontrés précédemment à certaines classes de groupes particulières, incluant les produits graphés, les groupes de diagrammes introduits par Guba et Sapir, certains produits en couronne, et certains graphes de groupes. Les produits graphés constituent notre application la plus naturelle, où le lien entre le groupe et son graphe quasi-médian associé est particulièrement fort et explicite; en particulier, nous sommes capables de déterminer précisément quand un produit graphé est relativement hyperboliqueThe class of quasi-median graphs is a generalisation of median graphs, or equivalently of CAT(0) cube complexes. The purpose of this thesis is to introduce these graphs in geometric group theory. In the first part of our work, we extend the definition of hyperplanes from CAT(0) cube complexes, and we show that the geometry of a quasi-median graph essentially reduces to the combinatorics of its hyperplanes. In the second part, we exploit the specific structure of the hyperplanes to state combination results. The main idea is that if a group acts in a suitable way on a quasi-median graph so that clique-stabilisers satisfy some non-positively curved property P, then the whole group must satisfy P as well. The properties we are interested in are mainly (relative) hyperbolicity, (equivariant) lp-compressions, CAT(0)-ness and cubicality. In the third part, we apply our general criteria to several classes of groups, including graph products, Guba and Sapir's diagram products, some wreath products, and some graphs of groups. Graph products are our most natural examples, where the link between the group and its quasi-median graph is particularly strong and explicit; in particular, we are able to determine precisely when a graph product is relatively hyperbolic
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Awang, Jennifer S. "Dots and lines : geometric semigroup theory and finite presentability." Thesis, University of St Andrews, 2015. http://hdl.handle.net/10023/6923.
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Geometric semigroup theory means different things to different people, but it is agreed that it involves associating a geometric structure to a semigroup and deducing properties of the semigroup based on that structure. One such property is finite presentability. In geometric group theory, the geometric structure of choice is the Cayley graph of the group. It is known that in group theory finite presentability is an invariant under quasi-isometry of Cayley graphs. We choose to associate a metric space to a semigroup based on a Cayley graph of that semigroup. This metric space is constructed by removing directions, multiple edges and loops from the Cayley graph. We call this a skeleton of the semigroup. We show that finite presentability of certain types of direct products, completely (0-)simple, and Clifford semigroups is preserved under isomorphism of skeletons. A major tool employed in this is the Švarc-Milnor Lemma. We present an example that shows that in general, finite presentability is not an invariant property under isomorphism of skeletons of semigroups, and in fact is not an invariant property under quasi-isometry of Cayley graphs for semigroups. We give several skeletons and describe fully the semigroups that can be associated to these.
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Gielen, Steffen C. M. "Geometric aspects of gauge and spacetime symmetries." Thesis, University of Cambridge, 2011. https://www.repository.cam.ac.uk/handle/1810/240578.
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We investigate several problems in relativity and particle physics where symmetries play a central role; in all cases geometric properties of Lie groups and their quotients are related to physical effects. The first part is concerned with symmetries in gravity. We apply the theory of Lie group deformations to isometry groups of exact solutions in general relativity, relating the algebraic properties of these groups to physical properties of the spacetimes. We then make group deformation local, generalising deformed special relativity (DSR) by describing gravity as a gauge theory of the de Sitter group. We find that in our construction Minkowski space has a connection with torsion; physical effects of torsion seem to rule out the proposed framework as a viable theory. A third chapter discusses a formulation of gravity as a topological BF theory with added linear constraints that reduce the symmetries of the topological theory to those of general relativity. We discretise our constructions and compare to a similar construction by Plebanski which uses quadratic constraints. In the second part we study CP violation in the electroweak sector of the standard model and certain extensions of it. We quantify fine-tuning in the observed magnitude of CP violation by determining a natural measure on the space of CKM matrices, a double quotient of SU(3), introducing different possible choices and comparing their predictions for CP violation. While one generically faces a fine-tuning problem, in the standard model the problem is removed by a measure that incorporates the observed quark masses, which suggests a close relation between a mass hierarchy and suppression of CP violation. Going beyond the standard model by adding a left-right symmetry spoils the result, leaving us to conclude that such additional symmetries appear less natural.
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Sisto, Alessandro. "Geometric and probabilistic aspects of groups with hyperbolic features." Thesis, University of Oxford, 2013. http://ora.ox.ac.uk/objects/uuid:bcf456c4-eef0-4fe8-bb7d-8b15f9cf7b18.
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The main objects of interest in this thesis are relatively hyperbolic groups. We will study some of their geometric properties, and we will be especially concerned with geometric properties of their boundaries, like linear connectedness, avoidability of parabolic points, etc. Exploiting such properties will allow us to construct, under suitable hypotheses, quasi-isometric embeddings of hyperbolic planes into relatively hyperbolic groups and quasi-isometric embeddings of relatively hyperbolic groups into products of trees. Both results have applications to fundamental groups of 3-manifolds. We will also study probabilistic properties of relatively hyperbolic groups and of groups containing ``hyperbolic directions'' despite not being relatively hyperbolic, like mapping class groups, Out(Fn), CAT(0) groups and subgroups of the above. In particular, we will show that the elements that generate the ``hyperbolic directions'' (hyperbolic elements in relatively hyperbolic groups, pseudo-Anosovs in mapping class groups, fully irreducible elements in Out(Fn) and rank one elements in CAT(0) groups) are generic in the corresponding groups (provided at least one exists, in the case of CAT(0) groups, or of proper subgroups). We also study how far a random path can stray from a geodesic in the context of relatively hyperbolic groups and mapping class groups, but also of groups acting on a relatively hyperbolic space. We will apply this, for example, to show properties of random triangles.
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Sercombe, Damian. "A family of uniform lattices acting on a Davis complex with a non-discrete set of covolumes." Thesis, The University of Sydney, 2015. http://hdl.handle.net/2123/16026.
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Let (W,S) be a Coxeter system with Davis complex Σ. The polyhedral automorphism group G of Σ is a locally compact group under the compact-open topology. If G is a discrete group (as characterised by Haglund-Paulin), then the set Vu(G) of uniform lattices in G is discrete. Whether the converse is true remains an open problem. Under certain assumptions on (W,S), we show that Vu(G) is non-discrete and contains rationals (in lowest form) with denominators divisible by arbitrarily large powers of any prime less than a fixed integer. We explicitly construct our lattices as fundamental groups of complexes of groups with universal cover Σ. We conclude with a new proof of an already known analogous result for regular right-angled buildings.
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Barrett, Benjamin James. "Detecting topological properties of boundaries of hyperbolic groups." Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/285572.
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In general, a finitely presented group can have very nasty properties, but many of these properties are avoided if the group is assumed to admit a nice action by isometries on a space with a negative curvature property, such as Gromov hyperbolicity. Such groups are surprisingly common: there is a sense in which a random group admits such an action, as do some groups of classical interest, such as fundamental groups of closed Riemannian manifolds with negative sectional curvature. If a group admits an action on a Gromov hyperbolic space then large scale properties of the space give useful invariants of the group. One particularly natural large scale property used in this way is the Gromov boundary. The Gromov boundary of a hyperbolic group is a compact metric space that is, in a sense, approximated by spheres of large radius in the Cayley graph of the group. The technical results contained in this thesis are effective versions of this statement: we see that the presence of a particular topological feature in the boundary of a hyperbolic group is determined by the geometry of balls in the Cayley graph of radius bounded above by some known upper bound, and is therefore algorithmically detectable. Using these technical results one can prove that certain properties of a group can be computed from its presentation. In particular, we show that there are algorithms that, when given a presentation for a one-ended hyperbolic group, compute Bowditch's canonical decomposition of that group and determine whether or not that group is virtually Fuchsian. The final chapter of this thesis studies the problem of detecting Cech cohomological features in boundaries of hyperbolic groups. Epstein asked whether there is an algorithm that computes the Cech cohomology of the boundary of a given hyperbolic group. We answer Epstein's question in the affirmative for a restricted class of hyperbolic groups: those that are fundamental groups of graphs of free groups with cyclic edge groups. We also prove the computability of the Cech cohomology of a space with some similar properties to the boundary of a hyperbolic group: Otal's decomposition space associated to a line pattern in a free group.
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Solomyak, Margarita. "Essential spanning forests and electric networks in groups /." Thesis, Connect to this title online; UW restricted, 1997. http://hdl.handle.net/1773/5767.
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Kuckuck, Benno. "Finiteness properties of fibre products." Thesis, University of Oxford, 2012. http://ora.ox.ac.uk/objects/uuid:a9624d17-9d11-4bd0-8c46-78cbba73469c.
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A group Γ is of type Fn for some n ≥ 1 if it has a classifying complex with finite n-skeleton. These properties generalise the classical notions of finite generation and finite presentability. We investigate the higher finiteness properties for fibre products of groups.
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Cornwell, Christopher R. "On the Combinatorics of Certain Garside Semigroups." Diss., CLICK HERE for online access, 2006. http://contentdm.lib.byu.edu/ETD/image/etd1381.pdf.
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Cannas, Sonia. "Geometric representation and algebraic formalization of musical structures." Thesis, Strasbourg, 2018. http://www.theses.fr/2018STRAD047/document.
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Cette thèse présente des généralisations u groupe néo-riemannien PLR, que agit sur l'ensemble des 24 triades majeures et mineures. Le travail commence par une reconstruction de l'histoire de Tonnetz, un graphe associé aux trois transformations qui génèrent le groupe PLR. La thèse présente deux généralisations du groupe PLR pour les accords de septième. Le premier agit sur le tournage des septièmes de dominantes, mineure, semi-diminuée, majeure et diminuée, le second comprend également la septième mineure majeur, majeure augmenté, l'augmentée et la septième dedominante bémol. Nous avons également classé les transformations les plus parcimonieuses parmi les 4 triades (majeure, mineure, augmentée et diminuée) et avons étudié le groupe généré par celles-ci. Enfin, nous avons introduit une approche générale permettant de définir des opérations parcimonieuses entre les accords de septième et de triade, mais aussi les opérations déjà connues entre triades et celles entre septièmesThis thesis presents a generalizations of the neo-Riemannian PLR-group, that acts on the set of 24 major and minor triads. The work begins with a reconstruction on the history of the Tonnetz, a graph associated with the three transformations that generate the PLR-group. The thesis presents two generalizations of the PLR-group for seventh chords. The first one acts on the set of dominant, minor, semi-diminished, major and diminished sevenths, the second one also includes minor major, augmented major, augmented, dominant seventh flat five. We considered the most parsimonious operations exchanging two types of sevenths, moving a single note by a semitone or a whole tone. We also classified the most parsimonious transformations among the 4 types of triads (major, minor,augmented and diminished) and studied the group generated by them. Finally, we have introduced a general approach to define parsimonious operations between sevenths and triads, but also the operations already known between triads and those between sevenths
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Woodruff, Benjamin M. "Statistical Properties of Thompson's Group and Random Pseudo Manifolds." Diss., CLICK HERE for online access, 2005. http://contentdm.lib.byu.edu/ETD/image/etd854.pdf.
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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.
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Cette thèse est consacrée à l'étude du groupe d'automorphismes de groupes agissant sur des arbres d'une part, et du rang des groupes de Coxeter d'autre part.Via 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
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Gardam, Giles. "Encoding and detecting properties in finitely presented groups." Thesis, University of Oxford, 2017. https://ora.ox.ac.uk/objects/uuid:0c8a7009-7e04-4f66-911b-298ad87061fb.
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In this thesis we study several properties of finitely presented groups, through the unifying paradigm of encoding sought-after group properties into presentations and detecting group properties from presentations, in the context of Geometric Group Theory. A group law is said to be detectable in power subgroups if, for all coprime m and n, a group G satisfies the law if and only if the power subgroups G(m) and G(n) both satisfy the law. We prove that for all positive integers c, nilpotency of class at most c is detectable in power subgroups, as is the k-Engel law for k at most 4. In contrast, detectability in power subgroups fails for solvability of given derived length: we construct a finite group W such that W(2) and W(3) are metabelian but W has derived length 3. We analyse the complexity of the detectability of commutativity in power subgroups, in terms of finite presentations that encode a proof of the result. We construct a census of two-generator one-relator groups of relator length at most 9, with complete determination of isomorphism type, and verify a conjecture regarding conditions under which such groups are automatic. Furthermore, we introduce a family of one-relator groups and classify which of them act properly cocompactly on complete CAT(0) spaces; the non-CAT(0) examples are counterexamples to a variation on the aforementioned conjecture. For a subclass, we establish automaticity, which is needed for the census. The deficiency of a group is the maximum over all presentations for that group of the number of generators minus the number of relators. Every finite group has non-positive deficiency. For every prime p we construct finite p-groups of arbitrary negative deficiency, and thereby complete Kotschick's proposed classification of the integers which are deficiencies of Kähler groups. We explore variations and embellishments of our basic construction, which require subtle Schur multiplier computations, and we investigate the conditions on inputs to the construction that are necessary for success. A well-known question asks whether any two non-isometric finite volume hyperbolic 3-manifolds are distinguished from each other by the finite quotients of their fundamental groups. At present, this has been proved only when one of the manifolds is a once-punctured torus bundle over the circle. We give substantial computational evidence in support of a positive answer, by showing that no two manifolds in the SnapPea census of 72 942 finite volume hyperbolic 3-manifolds have the same finite quotients. We determine examples of sizeable graphs, as required to construct finitely presented non-hyperbolic subgroups of hyperbolic groups, which have the fewest vertices possible modulo mild topological assumptions.
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Bounds, Jordan. "On the quasi-isometric rigidity of a class of right-angled Coxeter groups." Bowling Green State University / OhioLINK, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1561561078356503.
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Lynch, Keith. "On triangles and quadrilaterals of groups." Diss., Virginia Tech, 1995. http://hdl.handle.net/10919/38574.
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This dissertation demonstrates the existence of a pair of algebraic and geometric structures on triangles of groups and on quadrilaterals of groups. These structures are an automatic and biautomatic structure. In addition, this paper also discusses the growth function for the quadrilaterals.
We show that these groups have these desired structures and discuss what they are. We also give an extraordinary example of a pair of quadrilaterals of groups that are defined nearly identically but do not behave alike.Ph. D.
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Sale, Andrew W. "The length of conjugators in solvable groups and lattices of semisimple Lie groups." Thesis, University of Oxford, 2012. http://ora.ox.ac.uk/objects/uuid:ea21dab2-2da1-406a-bd4f-5457ab02a011.
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The conjugacy length function of a group Γ determines, for a given a pair of conjugate elements u,v ∈ Γ, an upper bound for the shortest γ in Γ such that uγ = γv, relative to the lengths of u and v. This thesis focuses on estimating the conjugacy length function in certain finitely generated groups. We first look at a collection of solvable groups. We see how the lamplighter groups have a linear conjugacy length function; we find a cubic upper bound for free solvable groups; for solvable Baumslag--Solitar groups it is linear, while for a larger family of abelian-by-cyclic groups we get either a linear or exponential upper bound; also we show that for certain polycyclic metabelian groups it is at most exponential. We also investigate how taking a wreath product effects conjugacy length, as well as other group extensions. The Magnus embedding is an important tool in the study of free solvable groups. It embeds a free solvable group into a wreath product of a free abelian group and a free solvable group of shorter derived length. Within this thesis we show that the Magnus embedding is a quasi-isometric embedding. This result is not only used for obtaining an upper bound on the conjugacy length function of free solvable groups, but also for giving a lower bound for their Lp compression exponents. Conjugacy length is also studied between certain types of elements in lattices of higher-rank semisimple real Lie groups. In particular we obtain linear upper bounds for the length of a conjugator from the ambient Lie group within certain families of real hyperbolic elements and unipotent elements. For the former we use the geometry of the associated symmetric space, while for the latter algebraic techniques are employed.
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Lima, Francismar Ferreira 1985. "Pontos fixos por grupos finitos agindo sobre grupos solúveis de tipo FP infinito." [s.n.], 2013. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306924.
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Orientador: Dessislava Hristova KochloukovaDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica
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Resumo: O resumo poderá ser visualizado no texto completo da tese digital
Abstract: The complete Abstract is available with the full electronic document.
Mestrado
Matematica
Mestre em Matemática
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Shim, Sangho. "Large scale group network optimization." Diss., Atlanta, Ga. : Georgia Institute of Technology, 2009. http://hdl.handle.net/1853/31737.
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Thesis (Ph.D)--Industrial and Systems Engineering, Georgia Institute of Technology, 2010.Committee Chair: Ellis L. Johnson; Committee Member: Brady Hunsaker; Committee Member: George Nemhauser; Committee Member: Jozef Siran; Committee Member: Shabbir Ahmed; Committee Member: William Cook. Part of the SMARTech Electronic Thesis and Dissertation Collection.
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Kelvey, Robert J. Kelvey. "Properties of groups acting on Twin-Trees and Chabauty space." Bowling Green State University / OhioLINK, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1479423366082688.
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Disarlo, Valentina. "Combinatorial methods in Teichmüller theory." Doctoral thesis, Scuola Normale Superiore, 2013. http://hdl.handle.net/11384/85687.
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In this thesis we deal with combinatorial and geometric properties of arc complexes and triangulation graphs, and we will provide some applications to the study of the mapping class group and to the Teichmüller theory of a bordered surface. The thesis is divided into two parts. In the former we deal with the problem of combinatorial rigidity of arc complexes. In the latter we study some large-scale properties of the arc complex and the 1-skeleton of its dual, the so-called ideal triangulation graph.
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Almeida, Kisnney Emiliano de 1984. "Sobre os sigma-invariantes unidimensionais de grupos de Artin." [s.n.], 2012. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306925.
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Orientador: Dessislava Hristova KochloukovaTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
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Resumo: A teoria de ?-invariantes surgiu do trabalho de Bieri e Strebel, que definiram o primeiro ?-invariante, apenas para grupos metabelianos, e o usaram para descrever os grupos metabelianos finitamente gerados [BiSt]. Posteriormente, foram definidos os ?m-invariantes homotópicos e homológicos de grupos finitamente gerados arbitrários [BiNSt]. Estes são certos subconjuntos da esfera de caracteres profundamente relacionados às propriedades de finitude Fm e FPm, respectivamente. Os grupos de Artin formam uma grande classe de grupos, cada um associado a um grafo rotulado, que inclui algumas subclasses importantes, como "Braid groups" e "Rightangled Artin groups"...Observação: O resumo, na íntegra, poderá ser visualizado no texto completo da tese digital
Abstract: The ?-invariants theory arose from the work of Bieri and Strebel, who defined the first ?-invariant, for metabelian groups only, and used it to describe the finitely presented metabelian groups [BiSt]. Later on, the homotopical and homological ?m-invariants of arbitrary finitely generated groups were defined [BiRe]. These are certain subsets of the sphere of characters deeply related to the finiteness properties Fm and FPm, respectively. The Artin groups form a large class of groups, each one associated to a labeled graph, that includes some important subclasses, as Braid groups and Right-angled Artin groups...Note: The complete abstract is available with the full electronic document
Doutorado
Matematica
Doutor em Matemática
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Yang, Ruotao. "Twisted Whittaker category on affine flags and category of representations of mixed quantum group." Electronic Thesis or Diss., Université de Lorraine, 2020. http://www.theses.fr/2020LORR0064.
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Supposons que G est un groupe reductif. Nous avons l’équivalence géométrique de Satake qui identifie Sph(G), les faisceau pervers G (O) équivalentes sur grassmannin affine en tant que catégorie de représentation dimensionnelle finie de H, le groupe duel de Langlands de G. Notez qu’il existe une équivalence : Whit(Gr) = Sph(G). Ici, Whit(Gr) est la catégorie de module D (N(K), \chi)-equivalent sur Gr. Maintenant, la catégorie de représentation admet une déformation par la catégorie des représentations de groupe quantique. Le côté gauche, nous pouvons considérer les D modules torsadé sur grassmannin affine. C'est l'équivalence locale fondamentale : Whit_q(Gr)= Rep_q(H) . Récemment, D. Gaitsgory a proposé sa version ramifiée. Nous considérons les drapeaux affines au lieu des grassmanniens affines. Dans ce cas, nous devons remplacer la catégorie des représentations de groupe quantique par une autre catégorie, la catégorie des représentations de groupe quantique mixte. Whit_q(Fl)= Rep_q^{mix}(H) . Nous prouvons que la catégorie de Whittaker torsadé sur la variété de drapeau affine et la catégorie de représentations du groupe quantique mixte sont équivalentesSuppose that G is a reductive group. We have the geometric Satake equivalence which identifies Sph (G), the perverse G (O) equivalent D-modules on affine grassmannin as the category of finite dimensional representation of H, the Langlands dual group of G. We note that: Whit(Gr) = Sph(G). Here, Whit (Gr) is the module category D (N (K), \ chi) -equivalent on Gr. Now, the category of representation admits a deformation by the category of representations of quantum group. On the Whittaker side, we can consider the twisted D-modules on affine grassmannin. This is the fundamental local equivalence: Whit_q (Gr) = Rep_q (H) . Recently, D. Gaitsgory proposed its ramified version. We consider the affine flags instead of the affine grassmannians. In this case, we have to replace the category of quantum group representations with another category, the category of mixed quantum group representations. Whit_q (Fl) = Rep_q ^ {mix} (H) . We prove that the category of twisted Whittaker D-modules on the affine flags and the category of representations of the mixed quantum group are equivalent
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Disarlo, Valentina. "Combinatorial methods in Teichmüller theory." Phd thesis, Université de Strasbourg, 2013. http://tel.archives-ouvertes.fr/tel-00875029.
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In this thesis we deal with combinatorial and geometric properties of arc complexes and triangulation graphs, and we will provide some applications to the study of the mapping class group and to the Teichmüller theory of a bordered surface. The thesis is divided into two parts. In the former we deal with the problem of combinatorial rigidity of arc complexes. In the latter we study some large-scale properties of the arc complex and the 1-skeleton of its dual, the so-called ideal triangulation graph.
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Maillot, Sylvain. "Quasi-isométries, groupes de surfaces et orbifolds fibrés de Seifert." Phd thesis, Université Paul Sabatier - Toulouse III, 2000. http://tel.archives-ouvertes.fr/tel-00001342.
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Le résultat principal est une caractérisation homotopique des orbifolds de dimension 3 qui sont fibrés de Seifert : si O est un orbifold de dimension 3 fermé, orientable et petit dont le groupe fondamental admet un sous-groupe infini cyclique normal, alors O est de Seifert. Ce théorème généralise un résultat de Scott, Mess, Tukia, Gabai et Casson-Jungreis pour les variétés. Il repose sur une caractérisation des groupes de surfaces virtuels comme groupes quasi-isométriques à un plan riemannien complet. D'autres résultats sur les quasi-isométries entre groupes et surfaces sont obtenus.
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Camelo, Botero Miguel Hernando. "A geometric routing scheme in word-metric spaces for data networks." Doctoral thesis, Universitat de Girona, 2014. http://hdl.handle.net/10803/283749.
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This research work explores the use of the Greedy Geometric Routing (GGR) schemes to solve the scalability problem of the routing systems in Internet-like networks and several families of Data Center architectures. We propose a novel and simple embedding of any connected finite graph into a Word-Metric space, i.e., a metric space generated by algebraic groups. Then, built on top of this greedy embedding, we propose three GGR schemes and we prove the theoretical upper bounds of the Routing Table size, vertex label size and stretch. The first scheme works for any kind of graph and the other two are specialized for Internet-like and several families of DC topologiesEste trabajo de investigación explora el uso de esquemas de Enrutamiento Geométrico Greedy (Greedy Geometric Routing o GGR) para resolver el problema de escalabilidad de los sistemas de encaminamiento de redes tipo Internet y de varias arquitecturas para Centros de Datos (Data Centers o DCs). Nosotros proponemos un nuevo y simple método de incrustación (embedding) de cualquier grafo finito y conectado en un espacio métrico de palabras (Word-Metric space), es decir, un espacio métrico generado por grupos algebraicos. Luego, construidos sobre esta incrustación, proponemos tres esquemas de GGR y derivamos los límites superiores teóricos de sus tablas de encaminamiento (Routing Table o RT), las etiquetas de los vértices y el stretch. El primer esquema trabaja sobre cualquier tipo de grafo y los otros dos son especializados para topologías tipo Internet y varias familias de arquitecturas de DCs
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D'Andrea, Joy. "Fundamental Transversals on the Complexes of Polyhedra." Scholar Commons, 2011. http://scholarcommons.usf.edu/etd/3746.
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We present a formal description of `Face Fundamental Transversals' on the faces of the Complexes of polyhedra (meaning threedimensional polytopes). A Complex of a polyhedron is the collection of the vertex points of the polyhedron, line segment edges and polygonal faces of the polyhedron. We will prove that for the faces of any 3-dimensional complex of a polyhedron under face adjacency relations, that a `Face Fundamental Transversal' exists, and it is a union of the connected orbits of faces that are intersected exactly once. While exploring the problem of finding a face fundamental transversal, we have found a partial result for edges that are incident to faces in a face fundamental transversal. Therefore we will present this partial result, as The Edge Transversal Proposition 1. We will also discuss a few conjectures that arose out this proposition. In order to reach our approaches we will first discuss some history of polyhedra, group theory, and incorporate a little crystallography, as this will appeal to various audiences.
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Silva, Leonardo de Amorin e. 1980. "Grupos abelianos-por-(nilpotentes de classe 2)." [s.n.], 2014. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306919.
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Orientador: Dessislava Hristova KochloukovaTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica
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Resumo: Nesta tese consideramos uma extensão cindida G de um grupo abeliano A por um grupo nilpotente (de classe 2) Q e provamos dois resultados. Primeiro, se Q age nilpotentemente sobre A e G tem tipo FP2, calculamos o sigma invariante de G em dimensão 2. Segundo, se G tem tipo FP4, mostramos que cada quociente de G tem tipo FP4
Abstract: In this thesis we consider a split extension G of an abelian group A by a nilpotent group (class 2) Q and prove two results. First, if Q acts nilpotently on A and G has type FP2, compute the sigma invariant of G in dimension 2. Second, if G has type FP4, we show that every quotient G has type FP4
Doutorado
Matematica
Doutora em Matemática
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Nicol, Andrew. "Quasi-isometries of graph manifolds do not preserve non-positive curvature." The Ohio State University, 2014. http://rave.ohiolink.edu/etdc/view?acc_num=osu1405894640.
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Dufloux, Laurent. "Dimension de Hausdorff des ensembles limites." Thesis, Sorbonne Paris Cité, 2015. http://www.theses.fr/2015USPCD022/document.
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Soit G le groupe SO°(1, n) (n ≥ 3) ou PU(1, n) (n ≥ 2) et fixons une décomposition d'Iwasawa G = KAN. Soit ɼ un sous-groupe discret de G, que nous supposons Zariski-dense et de mesure de Bowen-Margulis-Sullivan finie. Lorsque G = SO°(1, n), nous étudions la géométrie de la mesure de Bowen-Margulis-Sullivan le long des sous-groupes fermés connexes de N, en lien avec la dichotomie de Mohammadi-Oh. Nous établissons des résultats déterministes sur la dimension des projections de la mesure de Patterson- Sullivan. Lorsque G = PU(1, n), nous relions la géométrie de la mesure de Bowen- Margulis-Sullivan le long du centre du groupe de Heisenberg au problème du calcul de la dimension de Hausdorff de l'ensemble limite relativement à la distance sphérique au bord. Nous calculons cette dimension pour certains groupes de SchottkyLet G be the group SO° (1,n) (n ≥ 3) or PU(1, n) (n ≥ 2) and fix some Iwasawa decomposition G = KAN. Let ɼ be a discrete subgroup of G.We assume that ɼ is Zariski-dense with finite Bowen-Margulis-Sullivan measure. When G = SO°(1,n), we investigate the geometry of the Bowen-Margulis-Sullivan measure elong connected closed subgroups of N. This is related to the Mohammadi-Oh dichotomy. We then prove deterministic results on the dimension of projections of Patterson-Sullivan measure. When G = PU(1,n), we relate the geometry of Bowen-Margulis-Sullivan measure along the center of Heisenberg group to the problem of computing the Hausdorff dimension of the limit set with respect to the spherical metric on the boudary. We construct some Schottky subgroups for wich we are able to compute this dimension
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Petersen, Willis L. "The Lie Symmetries of a Few Classes of Harmonic Functions." Diss., CLICK HERE for online access, 2005. http://contentdm.lib.byu.edu/ETD/image/etd837.pdf.
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Bouette, Margot. "Sur la croissance des automorphismes des groupes de Baumslag-Soliltar." Thesis, Rennes 1, 2016. http://www.theses.fr/2016REN1S053/document.
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Un groupe de Baumslag-Solitar est un groupe dont la présentation est, pour p et q entiers non nuls. A chaque groupe de Baumslag-Solitar est associé un espace de déformation D p, q d'actions sur des arbres analogue à l'outre espace. Aut(BS(p, q)) agit sur cet espace ce qui induit une action du groupe des automorphismes extérieurs Out(BS(p,q)). Nous nous intéresserons au cas plus complexe où q est un multiple de p et dans un premier temps, nous démontrerons que tout automorphisme de BS(p, pn) est réductible ce qui signifie qu'il existe un BS(p,pn)-arbre T et une application laissant invariante un certain type de forêt. Ce résultat nous amènera à introduire un nouvel espace de déformation et une classification des automorphismes de BS(p, pn) en trois catégories : elliptique, parabolique ou hyperbolique. A l'aide de cette classification, nous démontrerons que tout automorphisme est à croissance soit polynomiale soit exponentielleA Baumslag-Solitar group is a group given by the group presentation, for p and q non-zero integers. For each Baumslag-Solitar group we consider a deformation space D p, q which is analogue of Culler-Vogtmann's Outer Space. The action of Aut(BS(p, q)) on D p, q induces an action of the outer automorphism group Out(BS(pq)). We will focus on the case where p divides q. Firstly, we will show that every automorphism of BS(p,pn) is reducible which means that we can find a BS(p,pn)-tree T and a map that leaves a certain type of subforest invariant. This result leads us to introduce a new deformation space and a classification of the automorphisms of BS(p,pn) in three types : elliptic, parabolic or hyperbolic. Using this classification, we will show that the growth of every automorphism of BS(p,pn) is exponential or polynomial
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Bouljihad, Mohamed. "Propriété (T) de Kazhdan relative à l'espace." Thesis, Lyon, 2016. http://www.theses.fr/2016LYSEN010/document.
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L'objet de cette thèse est l'étude de la propriété (T) relative à l'espace (ou rigidité au sens de Popa) d'actions de groupes dénombrables sur des espaces de probabilité standards préservant une mesure de probabilité (pmp). Ces dix dernières années, la propriété (T) relative à l'espace a permis de résoudre de nombreux problèmes dans le cadre de la théorie ergodique des actions de groupes et des algèbres de von Neumann. Néanmoins, certains aspects théoriques de cette notion restent largement mystérieux. Une question encore ouverte consiste à déterminer les groupes admettant une action libre ergodique pmp ayant la propriété (T) relative à l'espace. Nous montrons dans cette thèse que les groupes de type fini non-moyennables linéaires sur un corps de caractéristique nulle admettent une action ergodique pmp possédant cette propriété. Si le groupe est à radical résoluble trivial, l'action que nous construisons est aussi libre.Pour ce faire, nous commençons par étudier la stabilité de la propriété (T) relative à l'espace vis-à-vis de différentes constructions d'actions pmp : produit, restriction, co-induction, induction. Puis, nous donnons une caractérisation de la propriété (T) relative à l'espace dans le cas d'actions pmp sur un espace homogène G/Λ de groupe de Lie p-adique d'un sous-groupe dénombrable Γ du groupe des transformations affines de G stabilisant le réseau Λ. L'action de Γ sur G/Λ a la propriété (T) relative à l'espace si et seulement s'il n'existe pas de mesure de probabilité Γ-invariante sur l'espace projectif de l'algèbre de Lie de G. Par ailleurs, nous étudions le cas d'actions de groupes par automorphismes sur des nilvariétés définies par des graphes finisThe purpose of this thesis is to study the Kazhdan's property (T) relative to the space (also called rigidity in the sense of Popa) of probability measure preserving actions of countable groups on standard probability measure spaces (p.m.p.).This last decade, some problems in the theory of ergodic theory and von Neumann algebras were solved using the property (T) relative to the space. However, the theoretical aspects of its study remain largely mysterious. An open question asks which groups admit a p.m.p. free and ergodic action which has the property (T) relative to the space. We show in this dissertation that every finitely-generated non-amenable linear groups over a field of characteristic zero admits a p.m.p. ergodic action which has this property. If this group has trivial solvable radical, we prove that these actions can be chosen to be free.In order to obtain these results, we start by investigating natural questions concerning the stability of the property (T) relative to the space through standard constructions : products, restriction, co-induction, induction. Then, we give a criterion for the property (T) relative to the space to hold in the case of p.m.p. actions on homogeneous space G/ Λ of a p-adic Lie group for a countable subgroup Γ of affine transformations of G stabilizing the lattice Λ. The action of Γ on G/Λ has the property (T) relative to the space if and only if the induced action of Γ on the projective space of the Lie algebra of G admits no invariant probability measure.Moreover, we study the case of actions by automorphims on nilvarietes defined by finite graphs
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Vittone, Davide. "Submanifolds in Carnot groups." Doctoral thesis, Scuola Normale Superiore, 2006. http://hdl.handle.net/11384/85698.
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