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On Poicarés Uniformization TheoremBartolini, Gabriel January 2006 (has links)
<p>A compact Riemann surface can be realized as a quotient space $\mathcal{U}/\Gamma$, where $\mathcal{U}$ is the sphere $\Sigma$, the euclidian plane $\mathbb{C}$ or the hyperbolic plane $\mathcal{H}$ and $\Gamma$ is a discrete group of automorphisms. This induces a covering $p:\mathcal{U}\rightarrow\mathcal{U}/\Gamma$.</p><p>For each $\Gamma$ acting on $\mathcal{H}$ we have a polygon $P$ such that $\mathcal{H}$ is tesselated by $P$ under the actions of the elements of $\Gamma$. On the other hand if $P$ is a hyperbolic polygon with a side pairing satisfying certain conditions, then the group $\Gamma$ generated by the side pairing is discrete and $P$ tesselates $\mathcal{H}$ under $\Gamma$.</p>
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On Poicarés Uniformization TheoremBartolini, Gabriel January 2006 (has links)
A compact Riemann surface can be realized as a quotient space $\mathcal/\Gamma$, where $\mathcal$ is the sphere $\Sigma$, the euclidian plane $\mathbb$ or the hyperbolic plane $\mathcal$ and $\Gamma$ is a discrete group of automorphisms. This induces a covering $p:\mathcal\rightarrow\mathcal/\Gamma$. For each $\Gamma$ acting on $\mathcal$ we have a polygon $P$ such that $\mathcal$ is tesselated by $P$ under the actions of the elements of $\Gamma$. On the other hand if $P$ is a hyperbolic polygon with a side pairing satisfying certain conditions, then the group $\Gamma$ generated by the side pairing is discrete and $P$ tesselates $\mathcal$ under $\Gamma$.
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Das Gitterpunktproblem in der hyperbolischen EbeneThirase, Jan 01 November 2000 (has links)
Es wird sowohl das klassischen Kreisproblem als auch dessen Verallgemeinerung auf geometrisch endliche Fuchssche Gruppen betrachten. Insbesondere werden obere und untere Schranken für die jeweiligen Zählfunktionen angeben. Mit einem Computerprogramm wird die Zählfunktion einer Heckegruppe bestimmt und damit eine Abschätzung ihres Konvergenzexponenten gegeben.
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Polígono fundamental associado ao grupo gerador da superfície / Associate of fundamental polygon generator surfaceGabriel Filho, Luiz Carlos 24 February 2011 (has links)
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Previous issue date: 2011-02-24 / In this paper we study a class of polygons on the Poincare disk, known as canonical Fricke polygon that are fundamental polygon related to a Fuchsian group, generating a surface of genus g. We rely on Article by Linda Keen [15], considering the case where the genus g > 0. Moreover, in order to apply the procedure adopted by Keen, we calculate the cycles and found the relationship of groups related to the tiles of the type {24λ + 4, 4} and {24λ − 12, 4}, which were originally obtained the tiles {12η − 8, 4} and {12η − 12, 4} given by Oliveira in [19]. Then we use a procedure developed by Agustini [1] to display the matrices associated with pairing functions coming this way to the vertices of the polygon associated key. / Neste trabalho fazemos um estudo de uma classe polígonos no disco de Poincaré, conhecidos como polígonos canônicos de Fricke, que são polígonos fundamentais relacionados a um grupo fuchsiano, gerador de uma superfície de gênero g. Nos baseamos no artigo de Linda Keen [15], considerando o caso em que o gênero g > 0. Além disso, com o intuito de aplicar o procedimento adotado por Keen, calculamos os ciclos e encontramos as relações do grupo relacionado aos ladrilhamentos de tipo {24λ +4, 4} e {24λ − 12, 4}, que originalmente foram obtidos dos ladrilhamentos {12η − 8, 4} e {12η − 12, 4} apresentados por Oliveira em [19]. Em seguida fazemos uso de um procedimento desenvolvido por Agustini [1] para exibir as matrizes associadas às funções de emparelhamento chegando desta maneira aos vértices do polígono fundamental associado.
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Elementos da teoria de Teichmüller / Elements of the Teichmüller theoryVizarreta, Eber Daniel Chuño 23 February 2012 (has links)
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.
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Emparelhamento de arestas de polígonos gerados por grafos / Side-pairing of polygons generated by graphsSilva, Gheyza Ferreira da 24 February 2011 (has links)
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Previous issue date: 2011-02-24 / This work has as main objective the study of side-pairing patterns for hyperbolic polygons with 12g−6 edges and angles 2π/3 generated by trivalent graphs, in the case when the quotient of the hyperbolic plane by a Fuchsian group Γ (generated by the side-pairing of the polygon), H2/Γ , is a closed surface of genus g, g ≥ 2. So we did a study in case of g = 2, based on [10] and for the case of g = 3, based on [17]. In this work, we deduce two ways to get closed paths in the trivalent graphs cited in [10] and [17] and we contribute with exemples and results for cases of g > 3. Moreover, we find generalizations for some of these side-pairing patterns. / Este trabalho tem como objetivo principal o estudo de emparelhamentos de arestas para polígonos hiperbólicos com 12g − 6 arestas e ângulos iguais a 2π/3 gerados por meio de grafos trivalentes, no caso em que o quociente do plano hiperbólico por um grupo Fuchsiano Γ (gerado pelo emparelhamento do polígono), H2/Γ , é uma superfície fechada de gênero g, g ≥ 2. Assim, fizemos
um estudo para o caso de g = 2 baseado em [10] e para o caso de g = 3, baseado em [17]. Neste trabalho, nós deduzimos duas formas de obter os caminhos fechados nos grafos trivalentes citados em [10] e [17] e contribuímos com exemplos e resultados para casos em que g > 3. Além disso, encontramos generalizações para alguns desses emparelhamentos de arestas.
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Elementos da teoria de Teichmüller / Elements of the Teichmüller theoryEber Daniel Chuño Vizarreta 23 February 2012 (has links)
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.
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