In this dissertation we study lattices in quotient spaces. The basic quotient spaces are: (1) n-dimensional euclidean tori, obtained from quotient of Rn by discrete groups of isometries ge- nerated by linearly independent translations and (2) hyperbolic °at g-tori (tori of genus g ¸ 2), obtained from quotient of hyperbolic plane by fuchsian groups. In the euclidean environment, the considered lattices are provided of the additive group Z2; while in the hyperbolic case the studied lattices are the geometrically uniform and the cyclic ones. / Neste trabalho estudamos reticulados em espaços quocientes. Os espaços quocientes considerados foram: (1) toros euclidianos n-dimensionais, obtidos pelo quociente de Rn por grupos discretos de isometrias gerados por translações linearmente independentes e (2) g-toros planos hiperbólicos (g ¸ 2) ; obtidos pelo quociente do plano hiperbólico por grupos fuchsianos. No caso euclidiano, os reticulados considerados foram provenientes de Z2; enquanto que no caso hiperbólico os reticulados estudados foram os geometricamente uniformes e os cíclicos. / Mestre em Matemática

Orientadores: Reginaldo Palazzo Jr., Mercio Botelho Faria / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-08T06:25:10Z (GMT). No. of bitstreams: 1 Vieira_VandenbergLopes_D.pdf: 990187 bytes, checksum: 2212b8074f5503f78aa813ce4422cc4b (MD5) Previous issue date: 2007 / Resumo: Dentro do contexto de projetar sistema de comunicação digital em espaços homogêneos, em particular, em espaços hiperbólicos, é necessário estabelecer um procedimento sistemático para construção de reticulados O, como elemento base para construção de constelações de sinais geometricamente uniformes. E através desse procedimento que identificamos as estruturas algébrica e geométrica além de construir códigos geometricamente uniformes em espaços homogêneos. Propomos, a partir desses reticulados, a construção de grupos fuchsianos aritméticos Tp obtidos de tesselações hiperbólicas {p; q}, derivados de álgebras de divisão dos quaternios A sobre corpos de números K. Generalizamos o processo de identificação desses grupos em ordens dos quatérnios (reticulados hiperbólicos), associadas às constelações de sinais geometricamente uniformes, provenientes de grupos discretos. Esse procedimento permite rotular os sinais das constelações construídas por elementos de uma estrutura algébrica / Abstract: Within the context of digital communications system in homogeneous space in particular, in hyperbolic spaces, it is necessary to establish systematic procedure for the construction of lattices O ; as the basic entity for construction of eometrically uniforms signal constellations. By this procedure we identify the algebraic and geometric structures to construct geometrically uniforms codes in homogeneous spaces. We propose, from lattices, the construction of arithmetic fuchsian groups ¡p obtained by hyperbolic tessellations {p; q}, derived from division quaternion algebras A over numbers fields K. We generalize the process of identification of these groups in quaternion orders (hyperbolic lattices), which are associated with geometrically uniforms signal constellations, proceeding from discrete groups. This procedure allows us to realize the labelling of the signals belonging to such constellations by elements of an algebraic structure / Doutorado / Telecomunicações e Telemática / Doutor em Engenharia Elétrica

Nesta disertação estudamos algumas ferramentas básicas relacionadas aos espaços de Teichmüller. Introduzimos o espaço de Teichmüller de gênero g ≥ 1, denotado por Tg. O objetivo principal é construir as coordenadas de Fenchel-Nielsen ωG : Tg → R3g-3+ × R3g-3 para cada grafo trivalente marcado G. / In this dissertation we study some basic tools related to Teichmüller space. We introduce the Teichmüller space of genus g ≥ 1, denoted by Tg. The main goal is to construct the Fenchel-Nielsen coordinates ωG : Tg → R3g-3+ × R3g-3 to each marked cubic graph G.

Dans cette thèse, nous nous intéressons à des singularités particulières dans les variétés de caractères. Dans le premier chapitre, on justifie que les caractères de représentations irréductibles d'un groupe fuchsien vers un groupe de Lie complexe semi-simple forment une orbifolde. Le lieu orbifold (i.e. l'ensemble des points dont l'isotropie n'est pas triviale) est constitué des caractères de représentations exceptionnelles. Dans le second chapitre, nous décrivons précisément le lieu orbifold quand le groupe de Lie est le groupe projectif linéaire sur un espace vectoriel complexe dont la dimension est un nombre premier. Dans le troisième et le quatrième chapitre nous cherchons à classifier les groupes d'isotropies possibles à conjugaison près apparaissant quand le groupe de Lie est respectivement un quotient du groupe spécial linéaire pour un espace vectoriel complexe de dimension finie quelconque dans le troisième chapitre et un quotient du groupe de spin complexe dans le quatrième chapitre. / Ln this thesis, we want to understand some singularities in the character variety. ln a first chapter, we justify that the characters of irreducible representations from a Fuchsian group to a complex semi-simple Lie group is an orbifold. The orbifold locus is, then, the characters of bad representations. ln the second chapter, we focus on the case where the Lie group is the projectif linear group over a complex vector space whose dimension is a prime number. ln particular we give an explicit description of this locus. ln the third and fourth chapter, we describe the isotropy groups (i.e. the centralizers of bad subgroups) arising in the cases when the Lie group is a quotient of the special linear group of a complex vector space of finite dimension (third chapter) and when the Lie group is a quotient of a complex spin group in the fourth chapter.

The thesis is organized as follows: First we state basic ergodic theorems in Section 2 and introduce the notation of Cesàro averages for multiple operators in Section 3. We state a general theorem in Section 3 for groups that can be represented by a finite alphabet and a transition matrix. In the second part we show that finitely generated Fuchsian groups, with certain restrictions to the fundamental domain, admit such a representation. To develop the representation we give an introduction into Möbius transformations (Section 4), hyperbolic geometry (Section 5), the concept of Fuchsian groups and their action in the hyperbolic plane (Section 6) and fundamental domains (Section 7). As hyperbolic geometry calls for visualization we included images at various points to make the definitions and statements more approachable. With those tools at hand we can develop a geometrical coding for Fuchsian groups with respect to their fundamental domain in Section 8. Together with the coding we state in Section 9 the main theorem for Fuchsian groups. The last chapter (Section 10) is devoted to the application of the main theorem to three explicit examples. We apply the developed method to the free group F3, to a fundamental group of a compact manifold with genus two and we show why the main theorem does not hold for the modular group PSL(2, Z).:1 Introduction 2 Ergodic Theorems 2.1 Mean Ergodic Theorems 2.2 Pointwise Ergodic Theorems 2.3 The Limit in Ergodic Theorems 3 Cesàro Averages of Sphere Averages 3.1 Basic Notation 3.2 Cesàro Averages as Powers of an Operator 3.3 Convergence of Cesàro Averages 3.4 Invariance of the Limit 3.5 The Limit of Cesàro Averages 3.6 Ergodic Theorems for Strictly Markovian Groups 4 Möbius Transformations 4.1 Introduction and Properties 4.2 Classes of Möbius Transformations 5 Hyperbolic Geometry 5.1 Hyperbolic Metric 5.2 Upper Half Plane and Poincaré Disc 5.3 Topology 5.4 Geodesics 5.5 Geometry of Möbius Transformations 6 Fuchsian Groups and Hyperbolic Space 6.1 Discrete Groups 6.2 The Group PSL(2, R) 6.3 Fuchsian Group Actions on H 6.4 Fuchsian Group Actions on D 7 Geometry of Fuchsian Groups 7.1 Fundamental Domains 7.2 Dirichlet Domains 7.3 Locally Finite Fundamental Domains 7.3.1 Sides of Locally Finite Fundamental Domains 7.3.2 Side Pairings for Locally Finite Fundamental Domains 7.3.3 Finite Sided Fundamental Domains 7.4 Tessellations of Hyperbolic Space 7.5 Example Fundamental Domains 8 Coding for Fuchsian Groups 8.1 Geometric Alphabet 8.1.1 Alphabet Map 8.2 Transition Matrix 8.2.1 Irreducibility of the Transition Matrix 8.2.2 Strict Irreducibility of the Transition Matrix 9 Ergodic Theorem for Fuchsian Groups 10 Example Constructions 10.1 The Free Group with Three Generators 10.1.1 Transition Matrix 10.2 Example of a Surface Group 10.2.1 Irreducibility of the Transition Matrix 10.2.2 Strict Irreducibility of the Transition Matrix 10.3 Example of PSL(2, Z) 10.3.1 Irreducibility of the Transition Matrix 10.3.2 Strict Irreducibility of the Transition Matrix

Orientador: Reginaldo Palazzo Junior / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-04T17:37:04Z (GMT). No. of bitstreams: 1 Souza_MarioJosede_D.pdf: 1221200 bytes, checksum: f5bf0643e72a350fca4e873f9d0e91e2 (MD5) Previous issue date: 2005 / Resumo: Neste trabalho apresentamos uma linha de transmissão como uma modelagem hiperbólica; construímos constelações de sinais hiperbólicas a partir das tesselações regulares do tipo {12g - 6, 3}; estabelecemos um procedimento para a contagem do número de pontos (sinais) das constelações acima citadas e apresentamos as funções automorfas como um meio de trânsito entre o ambiente das linhas de transmissão (semiplano direito) e o ambiente das constelações construídas (as superfícies de Riemann) / Abstract: In this work we have introduced a transmission line as a hyperbolic modeling; we have constructed a signal constellation in the hyperbolic plane from regular tessellations such as the ones generated by {12g - 6, 3} ; we have established a procedure for couting the number of points of the constellations mentioned above. We have also presented the automorphic functions as a means of transit between the context of transmission line (right semiplane) and the context of the constellations which were built (Riemann's surfaces) / Doutorado / Telecomunicações e Telemática / Doutor em Engenharia Elétrica

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