In this work, we investigate the mid-range behavior of geometry along a grafting ray in Teichm\"{u}ller space. The main technique is to describe the hyperbolic metric $\sigma_{t}$ at a point along the grafting ray in terms of a conformal factor $g_{t}$ times the Thurston (grafted) metric and study solutions to the linearized Liouville equation. We give a formula that describes, at any point on a grafting ray, the change in length of a sum of distinguished curves in terms of the hyperbolic geometry at the point. We then make precise the idea that once the length of the grafting locus is small, local behavior of the geometry for grafting on a general manifold is like that of grafting on a cylinder. Finally, we prove that the sum of lengths of is eventually monotone decreasing along grafting rays.

In this thesis, we investigate the period mapping of Teichmuller space into the Siegel upper half space. This is constructed from integrals of a basis of holomorphic one-forms along closed curves of a basis of the Riemann surface.  We consider the Riemann, Teichmuller and Torelli moduli spaces and their representation in the Siegel upper half space, and its relation to orbits of a symplectic and a set of positive polarizations of a vector space of dimension equal to the genus of the surface. / October 2016

In this paper we consider the action of the mapping class group of a surface on the space of homomorphisms from the fundamental group of a surface into PSU(1,1). Goldman conjectured that when the surface is closed and of genus bigger than one, the action on non-Teichmuller connected components of the associated moduli space (i.e. the space of homomorphisms modulo conjugation) is ergodic. One approach to this question is to use sewing techniques which requires that one considers the action on the level of homomorphisms, and for surfaces with boundary. In this paper we consider the case of the one-holed torus with boundary condition, and we determine regions where the action is ergodic. This uses a combination of techniques developed by Goldman, and Pickrell and Xia. The basic result is an analogue of the result of Goldman's at the level of moduli.

<div>In the first part of the present work, we will study the harmonic maps onto Teichm\"uller space. We will formulate a general Bochner type formula for harmonic maps into Teichm\"uller space. We will also prove the existence theorem of equivariant harmonic maps from a symmetric space with finite volume into its Weil-Petersson completion $\overline{\mathcal{T}}$, by deforming an almost finite energy map in the sense of Saper into a finite energy map.</div><div><br></div><div>In the second part of the work, we discuss the superrigidity of mapping class group. We will provide a geometric proof of both the high rank and the rank one superrigidity of mapping class groups due to Farb-Masur and Yeung. </div>

Orientador: Alexandre Ananin / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-16T08:38:27Z (GMT). No. of bitstreams: 1 Goncalves_EduardoCarvalhoBento_M.pdf: 2390910 bytes, checksum: fac0be229bed582ba6336532e52d2cd1 (MD5) Previous issue date: 2010 / Resumo: Primeiramente, apresentamos uma introdução à geometria hiperbólica plana que pode ser útil, inclusive, para um principiante. A seguir, utilizando o conceito de "terremoto simples", descrevemos explicitamente, em termos de algumas coordenadas naturais, o espaço de Teichmüller T Hn de superfícies hiperelípticas. Esta descrição resulta simples: T Hn é o espaço de determinadas (2n ? 6)-uplas de pontos no bordo ideal do plano hiperbólico. Partindo da descrição em questão, diversos resultados são apresentados, incluindo: um critério simples e efetivo que permite verificar se uma dada representação de um grupo de superfície no grupo de isometrias do plano hiperbólico é fiel e discreta; uma demonstração nova e elementar de um resultado de W. Goldman caracterizando as representações fiéis e discretas como aquelas que têm invariante de Toledo maximal; uma demonstração nova e elementar de um teorema de D. Toledo referente à rigidez de representações de grupos de superfície no grupo de isometrias holomorfas do espaço hiperbólico complexo / Abstract: First, we present an introduction to plane hyperbolic geometry, which may be useful even for a beginner. Next, using the concept of "simple earthquake", we explicitly describe, in terms of some natural coordinates, the Teichmüller space T Hn of hyperelliptic surfaces. This description turns out to be simple: T Hn is the space of certain (2n ? 6)-tuples of points in the ideal boundary of the hyperbolic plane. Based on the description in question, many results are presented, including: a simple and effective criterion which allows one to verify if a given representation of a surface group in the group of isometries of the hyperbolic plane is faithful and discrete; a new and elementary proof for a result of W. Goldman, which characterizes the faithful and discrete representations as being those which have maximal Toledo invariant; a new and elementary proof for a theorem of D. Toledo, relative to the rigidity of representations of surface groups in the group of holomorphic isometries of the complex hyperbolic space. key-words: Area, discreteness, representations, plane hyperbolic geometry, Teichmüller space, complex hyperbolic geometry / Mestrado / Geometria / Mestre em Matemática

Orientador : Marcelo Firer / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-04T02:48:30Z (GMT). No. of bitstreams: 1 Faria_MercioBotelho_D.pdf: 4443274 bytes, checksum: 86dda25654f7eb724f654b696016fcf1 (MD5) Previous issue date: 2005 / Resumo: Este trabalho busca elementos para se determinar a densidade de empacotamento de esferas definida por reticulados no plano hiperbólico.Consideramos o espaço de teichmuller Tu de todas as superfícies orientadas com-pactas e fechadas de gênero 9 2: 2, as quais tem o plano hiperbólico como recobrimento universal riemanniano. É conhecido o sistema de coordenadas Fricke em Tu que associa a cada superfície um domínio fundamental de Voronoi-Dirichlet dado por um polígono convexo com 4g arestas. Sabemos que, fixado o gênero, a densidade cresce com o número de arestas do domínio de Voronoi-Dirichlet escolhido, de modo que é natural a busca por polígonos com um número máximo de arestas associado ao gênero dado, que é sempre limitado por 12g - 6.Neste trabalho, determinamos as coordenadas Fricke em Tu que associa a cada su-perfície um domínio de Voronoi-Dirichlet com 4g + 2 e 12g - 6 arestas. Além disso, determinamos e implementamos algoritmos para a determinação dos círculos inscrito e circunscrito de um polígono (em superfícies de curvatura constante). Estes algorit-mos, em sua generalidade tem complexidade O (n4) mas, restringindo os polígonos a vizinhanças abertas de um polígono dado, possui complexidade O (n), situação ótima.A determinação dos domínios de Voronoi-Dirichlet e dos círculos inscritos permitem definir a densidade de empacotamento diretamente nos espaços de teichmuller através de um sistema de equações polinomiais / Abstract: This work searches elements to determine the packing density of spheres defined by lattices in the hyperbolic plane. We consider the teichmüller space Tg of all closed compacts oriented surfaces of genus 9 ~ 2, which has the hyperbolic plane as universal covering rienmannian surface. It is known that the system of Fricke coordinates in Tg associates each surface to a fundamental of Voronoi-Dirichlet domain, given by convex polygon with 49 edges. We know that, with fixed genus, the density increases with the number of edges of the chosen Voronoi-Dirichlet domain. Thus it is naturallooking for polygons with a maximum number of edges associated to a given genus, which is always limited by 129 - 6.In this work, we determine Fricke coordinates in Tg which associates each surface to a Voronoi-Dirichlet domain with 49 + 2 and 129 - 6 edges. Furthermore, we determine and we program the algorithms for determination of the inscribed and circumscribed circles of a polygon (in surfaces of constant curvature). These algorithms, have com-plexity O (n4) , but when restricted to open neighbourhoods of a given polygon, have complexity O (n), best situation.The determination of the Voronoi-Dirichlet domain from the inscribed circles per-mits to define the packing of density directly on teichmüller spaces through a polyno-mials of system equations / Doutorado / Matematica / Doutor em Matemática

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18049号 / 理博第3927号 / 新制||理||1566(附属図書館) / 30907 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 望月 新一, 教授 玉川 安騎男, 講師 星 裕一郎 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20158号 / 理博第4243号 / 新制||理||1610(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 望月 新一, 教授 岡本 久, 教授 玉川 安騎男 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18769号 / 理博第4027号 / 新制||理||1580(附属図書館) / 31720 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 玉川 安騎男, 教授 小野 薫, 教授 望月 新一 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM