This thesis investigates the geometric and topological constraints placed on the quotient space of a Fuchsian or Kleinian group by requiring that the group admits a fundamental domain which is simultaneously a Ford domain and a Dirichlet domain. In the case of Fuchsian groups, a direct correspondence with reflection groups is proved, and this result is used to first find explicitly the 23 non-cocompact arithmetic maximal hyperbolic reflection groups in the group of isometries of the hyperbolic plane, and subsequently to test whether these groups are all congruence. In the case of Kleinian groups, similar results are shown, and some examples of reflection groups are considered. / text

<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>

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 $\Gamma$.

Siegel détermina un domaine fondamental à l'aide de la réduction de Minkowski des formes quadratiques. Il donna tous les détails concernant ce domaine pour le genre 1. C'est la détermination du domaine fondamental de Minkowski présentée comme deuxième condition et la condition maximal height présentée comme troisième condition, qui empêchent la précision exacte de ce domaine pour le cas général. Les derniers résultats ont été obtenus par Gottschling pour le genre 2 en 1959. Elle est depuis restée inexplorée et mal comprise notamment les différents domaines de Minkowski. Afin d'identifier ce domaine fondamental pour le genre 3, nous présentons des résultats concernant sa troisième condition. Chaque fonction abélienne peut être écrite en termes de fonctions rationnelles des fonctions thêta et de leurs dérivées. Cela permet l'expression de la solution des systèmes intégrables en fonction des fonctions thêta. Ces solutions sont pertinentes dans la description de surfaces de vagues d'eau, de l'optique non linéaire. Deconinck et Van Hoeij ont éveloppé et mis en oeuvre des algorithmes pour construire la matrice de Riemann et Deconinck et al. ont développé le calcul des fonctions thêta correspondantes. Deconinck et al. ont utilisé l'algorithme de Siegel pour atteindre approximativement le domaine fondamental de Siegel et ont adopté l'algorithme LLL pour trouver le vecteur le plus court. Alors que nous utilisons ici un nouvel algorithme de réduction de Minkowski jusqu'à dimension 5 et une détermination exacte du vecteur le plus court pour des dimensions supérieures. / Siegel determined a fundamental domain using the Minkowski reduction of quadratic forms. He gave all the details concerning this domain for genus 1. It is the determination of the Minkowski fundamental domain presented as the second condition and the maximal height condition, presented as the third condition, which prevents the exact determination of this domain for the general case. The latest results were obtained by Gottschling for the genus 2 in 1959. It has since remained unexplored and poorly understood, in particular the different regions of Minkowski reduction. In order to identify Siegel's fundamental domain for genus 3, we present some results concerning the third condition of this domain. Every abelian function can be written in terms of rational functions of theta functions and their derivatives. This allows the expression of solutions of integrable systems in terms of theta functions. Such solutions are relevant in the description of surface water waves, non linear optics. Because of these applications, Deconinck and Van Hoeij have developed and implemented al-gorithms for computing the Riemann matrix and Deconinck et al. have developed the computation of the corresponding theta functions. Deconinck et al. have used Siegel's algorithm to approximately reach the Siegel fundamental domain and have adopted the LLL reduction algorithm to nd the shortest lattice vector. However, we opt here to use a Minkowski algorithmup to dimension 5 and an exact determination of the shortest lattice vector for greater dimensions.

Dada uma ação isométrica livre do grupo binário tetraedral G sobre esferas de dimensão ímpar, obtemos uma decomposição celular finita explícita para as formas espaciais esféricas tetraedrais, fazendo uso do conceito de região (ou domínio) fundamental. A estrutura celular deixa explícita uma descrição do complexo de cadeias sobre o grupo G. Como aplicações, utilizamos o complexo de cadeias e a interpretação geométrica do produto cup para calcular o anel de cohomologia da forma espacial esférica tetraedral em dimensão três, e também calculamos a torção de Reidemeister destes espaços para uma determinada representação de G / Given a free isometric action of a binary tetrahedral group G on odd dimensional spheres, we obtain an explicit finite cellular decomposition of the tetrahedral spherical space forms, using the concept of fundamental domain. The cellular structure gives an explicit description of the associated cellular chain complex over the group G. As applications we use the chain complex and the geometric interpretation of the cup product to calculate the cohomology ring of the tetrahedral spherical space form in three dimension, and also compute the Reidemeister torsion of these spaces for a determined representation of G

Dada uma ação isométrica livre do grupo binário tetraedral G sobre esferas de dimensão ímpar, obtemos uma decomposição celular finita explícita para as formas espaciais esféricas tetraedrais, fazendo uso do conceito de região (ou domínio) fundamental. A estrutura celular deixa explícita uma descrição do complexo de cadeias sobre o grupo G. Como aplicações, utilizamos o complexo de cadeias e a interpretação geométrica do produto cup para calcular o anel de cohomologia da forma espacial esférica tetraedral em dimensão três, e também calculamos a torção de Reidemeister destes espaços para uma determinada representação de G / Given a free isometric action of a binary tetrahedral group G on odd dimensional spheres, we obtain an explicit finite cellular decomposition of the tetrahedral spherical space forms, using the concept of fundamental domain. The cellular structure gives an explicit description of the associated cellular chain complex over the group G. As applications we use the chain complex and the geometric interpretation of the cup product to calculate the cohomology ring of the tetrahedral spherical space form in three dimension, and also compute the Reidemeister torsion of these spaces for a determined representation of G

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