[pt] O principal objetivo deste trabalho é o estudo de representações que preservam tipo rho:Gamma - PU(2,1) de grupos triangulares Gamma no grupo de isometrias holomorfas do espaço hiperbólico complexo de dimensão dois H2C. O grupo triangular Gamma(p,q,r) é o grupo gerado por reflexões nos lados de um triângulo geodésico, com ângulos pi/p, pi/q e pi/r, no plano hiperbólico. Neste trabalho, nossas atenções são voltadas para os grupos Gamma (4,4,infinito) e Gamma(4,infinito,infinito). Demonstramos, entre outros resultados: Para cada caso, existe um caminho contínuo de representações rho_t que contém todas as representações que preservam tipo de Gamma em PU(2,1). Portanto, isto nos dá, em cada caso, uma descrição completa do espaço de representações de Gamma em PU(2,1). Para cada caso, existe um intervalo fechado J tal que rho_t é uma representação discreta e fiel se, e somente se, t pertence a J. Em cada caso, existe, na fronteira do espaço de deformações, uma representação com elementos parabólicos acidentais. Para demonstrar estes resultados, construímos parametrizações especiais de triângulos em H2C. Construímos poliedros fundamentais para os grupos e utilizamos uma variante do Teorema do Poliedro de Poincaré. / [en] The main aim of this work is to study type-preserving representations p: gamma PU(2, 1) of triangle groups _ in the group of holomorphic isometries of the twodimensional complex hyperbolic space H2C. The triangle group gamma(p, q, r) is the group generated by reflections in the sides of a geodesic triangle having angles pi/p, pi/q and pi/r. We focus our attention on the groups gamma(4,4, infinit) and gamma (4,infinit, infinit). Among other results, we prove that for each case: 1. There is a continuous path of representations pt which contains all type-preserving representations of gamma in PU (2,1) up to conjugation by isometries. This gives us a complete description of the representation space of gamma in PU(2,1). 2. There is a closed interval J such that pt is a discrete and faithful representation if and only if t belongs J. 3. On the boundary of the representation space there is a representation with accidental parabolic elements. To prove these results we give special parametrizations of triangles in H2C. We also build fundamental polyhedra for the groups and use a kind of Poincares Polyhedron Theorem.

We obtain explicit formulae for Lagrangian representations of the (p, q, r)-triangle group into the group of direct isometries of the complex hyperbolic plane in the case where p=q=r. Numerically approximated matrix generators of representations of the (p, p, p)-triangle group are obtained using a special basis. The numerical approximations are then used to guess the exact generators by a process utilizing the LLL algorithm. The matrices are proved rigorously to generate Lagrangian representations of the (p, p, p)-triangle group and are applied to the problem of deciding whether or not an interval contains representations of the (p, p, p)-triangle group which are not discrete.

We study relations between special elliptic isometries in the complex hyperbolic plane. A special elliptic isometry can be seen as a rotation around a fixed axis (a complex geodesic). Such an isometry is determined by specifying a nonisotropic point p (the polar point to the fixed axis) and a unitary complex number a, the angle of the isometry. Any relation between special elliptic isometries with rational angles gives rise to a representation H(k1;:::;kn) → PU(2;1), where H(k1;:::;kn) : = ⟨ r1; : : : ; rn ∣ rn : : : r1> = 1; rkii = 1 ⟩ and PU(2;1) stands for the group of orientation-preserving isometries of the complex hyperbolic plane. We denote by Rpα the special elliptic isometry determined by the nonisotropic point p and by the unitary complex number α. Relations of the form Rpnαn : : :Rp1α1 = 1 in PU(2;1), called special elliptic n-gons, can be modified by short relations known as bendings: given a product RqβRpα, there exists a one-parameter subgroup B : R → SU(2;1) such that B(s) is in the centralizer of Rqβ Rpα and RB(s)qβRB(s)pα = RqβRB(s)pα for every s ∈ R. Then, for each i = 1,...,n-1, we can change Rpi+1αi+1Rpiαi by RB(s)pi+1αi+1RB(s)piαi obtaining a new n-gon. We prove that the generic part of the space of pentagons with fixed angles and signs of points is connected by means of bendings. Furthermore, we describe certain length 4 relations, called f -bendings, and prove that the space of pentagons with fixed product of angles is connected by means of bendings and f -bendings. / Neste trabalho, estudamos relações entre isometrias elípticas especiais no plano hiperbólico complexo. Uma isometria elíptica especial pode ser vista como uma rotação em torno de um eixo fixo (uma geodésica complexa). Tal isometria é determinada especificando-se um ponto não-isotrópico p (o ponto polar do eixo fixo) bem como um número complexo unitário a (o ângulo da isometria). Qualquer relação entre isometrias elípticas especiais com ângulos racionais dá origem a uma representação H(k1;:::;kn) → PU(2;1), onde H(k1;:::;kn) : = ⟨ r1; : : : ; rn ∣ rn : : : r1 = 1; rkii = 1 ⟩ e PU(2;1) é o grupo de isometrias que preservam a orientação do plano hiperbólico complexo. Denotamos por Rpα a isometria elíptica especial determinada pelo ponto não-isotrópico p e pelo complexo unitário α. Relações da forma Rpnαn : : :Rp1α1 = 1 em PU(2;1), chamadas n-ágonos elípticos especiais, podem ser modificadas a partir de relações curtas conhecidas como bendings: dado um produto RqβRpα, existe um subgrupo uniparamétrico B : R → SU(2;1) tal que B(s) está no centralizador de RqβRpα e RB(s)qβRB(s)pα = RqβRpα para todo s ∈ R. Assim, para cada i = 1; : : : ;n-1, podemos mudar Rpi+1α+1Rpiαi por RB(s)pi+1α+1RB(s)piα+1RB(s)piαi obtendo um novo n-ágono. Provamos que a parte genérica do espaço de pentágonos com ângulos e sinais de pontos fixados é conexa por meio de bendings. Além disso, descrevemos certas relações de comprimento 4, os f -bendings, e provamos que o espaço de pentágonos com produto de ângulos fixado é conexo por meio de bendings e f -bendings.

We study relations between special elliptic isometries in the complex hyperbolic plane. A special elliptic isometry can be seen as a rotation around a fixed axis (a complex geodesic). Such an isometry is determined by specifying a nonisotropic point p (the polar point to the fixed axis) and a unitary complex number a, the angle of the isometry. Any relation between special elliptic isometries with rational angles gives rise to a representation H(k1;:::;kn) → PU(2;1), where H(k1;:::;kn) : = ⟨ r1; : : : ; rn ∣ rn : : : r1> = 1; rkii = 1 ⟩ and PU(2;1) stands for the group of orientation-preserving isometries of the complex hyperbolic plane. We denote by Rpα the special elliptic isometry determined by the nonisotropic point p and by the unitary complex number α. Relations of the form Rpnαn : : :Rp1α1 = 1 in PU(2;1), called special elliptic n-gons, can be modified by short relations known as bendings: given a product RqβRpα, there exists a one-parameter subgroup B : R → SU(2;1) such that B(s) is in the centralizer of Rqβ Rpα and RB(s)qβRB(s)pα = RqβRB(s)pα for every s ∈ R. Then, for each i = 1,...,n-1, we can change Rpi+1αi+1Rpiαi by RB(s)pi+1αi+1RB(s)piαi obtaining a new n-gon. We prove that the generic part of the space of pentagons with fixed angles and signs of points is connected by means of bendings. Furthermore, we describe certain length 4 relations, called f -bendings, and prove that the space of pentagons with fixed product of angles is connected by means of bendings and f -bendings. / Neste trabalho, estudamos relações entre isometrias elípticas especiais no plano hiperbólico complexo. Uma isometria elíptica especial pode ser vista como uma rotação em torno de um eixo fixo (uma geodésica complexa). Tal isometria é determinada especificando-se um ponto não-isotrópico p (o ponto polar do eixo fixo) bem como um número complexo unitário a (o ângulo da isometria). Qualquer relação entre isometrias elípticas especiais com ângulos racionais dá origem a uma representação H(k1;:::;kn) → PU(2;1), onde H(k1;:::;kn) : = ⟨ r1; : : : ; rn ∣ rn : : : r1 = 1; rkii = 1 ⟩ e PU(2;1) é o grupo de isometrias que preservam a orientação do plano hiperbólico complexo. Denotamos por Rpα a isometria elíptica especial determinada pelo ponto não-isotrópico p e pelo complexo unitário α. Relações da forma Rpnαn : : :Rp1α1 = 1 em PU(2;1), chamadas n-ágonos elípticos especiais, podem ser modificadas a partir de relações curtas conhecidas como bendings: dado um produto RqβRpα, existe um subgrupo uniparamétrico B : R → SU(2;1) tal que B(s) está no centralizador de RqβRpα e RB(s)qβRB(s)pα = RqβRpα para todo s ∈ R. Assim, para cada i = 1; : : : ;n-1, podemos mudar Rpi+1α+1Rpiαi por RB(s)pi+1α+1RB(s)piα+1RB(s)piαi obtendo um novo n-ágono. Provamos que a parte genérica do espaço de pentágonos com ângulos e sinais de pontos fixados é conexa por meio de bendings. Além disso, descrevemos certas relações de comprimento 4, os f -bendings, e provamos que o espaço de pentágonos com produto de ângulos fixado é conexo por meio de bendings e f -bendings.

Dans cette thèse, on s'intéresse à la construction et à la déformation de structures CR-sphériques sur des variétés de dimension 3. Pour le faire, on étudie en détail l'espace hyperbolique complexe, son groupe d'isométries et des objets géométriques liés à cet espace. On montre un théorème de chirurgie qui permet de construire des structures CR-sphériques sur des chirurgies de Dehn d'une variété à pointe portant une structure CR-sphérique : il s'applique aux structures de Deraux-Falbel sur le complémentaire du noeud de huit et à celles de Schwartz et de Parker-Will sur le complémentaire de l'entrelacs de Whitehead. On définit aussi les variétés de caractères de groupes de type fini pour les formes réelles de SL(n,C) comme des sous-ensembles de la variété des caractères SL(n,C) fixes par des involutions anti-holomorphes. Ces variétés de caractères, dont on étudie en détail l'exemple du groupe Z/3Z*Z/3Z, fournissent des espaces de déformation pour des représentations d'holonomie de structures CR-sphériques. À l'aide de ces espaces de déformations, et des outils liés aux sphères visuelles dans CP^2, on construit une déformation explicite du domaine de Ford construit par Parker et Will et qui donne une uniformisation CR-sphérique sur le complémentaire de l'entrelacs de Whitehead. Cette déformation fournit une infinité d'uniformisations CR-sphériques sur une chirurgie de Dehn particulière de cette variété, et des uniformisations CR-sphériques sur une infinité de chirurgies de Dehn sur le complémentaire de l'entrelacs de Whitehead. / In this thesis, we study the construction and deformation of spherical-CR structures on three dimensional manifolds. In order to do it, we give a detailed description of the complex hyperbolic plane, its group of isometries and some geometric objects attached to this space such as bisectors and extors. We show a surgery theorem which allows to construct spherical-CR on Dehn surgeries of a cusped spherical-CR manifold : this theorem can be applied for the Deraux-Falbel structure on the figure eight knot complement and for Schwartz's and Parker-Will structures on the Whitehead link complement. We also define the character varieties for a real form of SL(n,C) for finitely generated groups as some subsets of the SL(n,C)-character variety invariant under an anti-holomorphic involution. We study in detail the example of the group Z/3Z*Z/3Z. These character varieties give deformation spaces for the holonomy representations of spherical-CR structures. With these deformation spaces and tools related to the visual spheres of a point in CP^2, we construct an explicit deformation of the Ford domain constructed by Parker and Will, which gives a spherical-CR uniformisation of the Whitehead link complement. This deformation provides infinitely many spherical-CR uniformisations of a particular Dehn surgery of the manifold, and spherical-CR unifomisations for infinitely many Dehn surgeries of the Whitehead link complement.

Cette thèse s'intéresse à des aspects divers des structures affines complexes branchées sur les surfaces de Riemann.Dans une première partie, nous étudions un invariant algébrique de ces structures appelé holonomie, qui est une représentation du groupe fondamental de la surface sous-jacente dans le groupe affine. Nous démontrons un théorème caractérisant les représentations se réalisant comme l'holonomie d'une structure affine.Nous nous intéressons ensuite à la géométrie de certains espaces de modules de telles structures qui viennent naturellement avec une structure hyperbolique complexe. Nous décrivons cette géométrie en terme de dégénérescences de structures affines.Enfin, nous regardons une sous-classe de structures affines dont chaque élément induit une famille de feuilletages sur la surface sous-jacente. Nous relions ces feuilletages à des systèmes dynamiques unidimensionnels appelés échanges d'intervalles affines et nous étudions un cas particulier en détails. / This thesis deals with several aspects of branched, complex affine structures on Riemann surfaces.In a first chapter, we study an algebraic invariant of these structures called holonomy, which is a representation of the fundamental group of the underlying surface into the affine group. We prove a theorem characterising such representations that arise as the holonomy of an affine structure.In a second part, we study certain moduli spaces of affine tori which happen to have an additional complex hyperbolic structure. We analyse the geometry of this structures in terms of degenerations of the underlying affine tori.Finally, we narrow our interest to a subclass of affine structures each element of which inducing a family of foliations on the underlying topological surface. We link these foliations to 1-dimensional dynamical systems called affine interval exchange transformations and study a particular case in details.