NOT EUCLIDEAN GEOMETRIES

Author: ÁGUEDA MATÉ RAQUEL.
Year: 2005.
University: NACIONAL DE EDUCACIÓN A DISTANCIA.
Place of defense: UNED.
Place of preparation: FACULTAD DE CIENCIAS.

Summary: The objective of this thesis is to identify the area of deformation A group Kleinianos free and geographically finite T, generated by a parabolic and other loxodrómico, which correspond to discrete and faithful representations in PSL (2, C) of a core group 3-range M hyperbolic whose border is a bull with two perforations (from the strangle a curve no-divisora in a solid bull gender 2). The space of parameters studied are at the frontier of space Schottky gender 2. The idea is to use varieties deplisado, the place where the geometric R border convex kernel of the variety is plisada along a rolling geodetic fixed. We focus our interest in the event that this is a system of rolling curves. The main findings are as follows: * Parametrizamos a family of representations 1 (M) and get a coarse space for the parameters. * Being compressible OM, there are systems that curves are not homótopos on the surface, but we are in the range. We find a representative of each class of homotopía systems curves M, the only exception homotopía in OM. Its coordinates (parameterized class homotopía systems curves in OM given by Keen-Parker-Series) verify certain conditions. * We studied what possible systems of curves that can provide the location of geometric Pleated. * We estimate the terms of a greater degree of the polynomial of a trace element corresponding to a closed curve and simple. * Extend the work of L. Keen and C. Series: studying one-dimensional slices in OR, we see that the formula for the cup gained widespread those found for incrustamiento of Maskit for the bull with a slice of drilling and Riley for Dial four holes.

Author: GARCÍA HERAS JOSÉ LUIS.
Year: 2005.
University: NACIONAL DE EDUCACIÓN A DISTANCIA.
Place of defense: FACULTAD DE CIENCIAS.
Place of preparation: FACULTAD DE CIENCIAS.

Summary: This report examines the Hyperbolic geometry of polygons with any number of sides. In the literature on the subject, appear explicit formulas for polygons of right angles and, under certain conditions, for other estates with a low sides. It addresses the problem in general: given a collection of angles, what conditions must meet the lengths of the sides for a polygon with those angles, and in fact there is convex. They also are various formulas that allow explicit calculate distances between any two sides of the site, between one side and a vertex outside him and between any two vertices, which allows for independent relationship between elements of a polygon Hyperbolic anyone. In addition, through the expressions obtained from the matrix and vertex normal vectors at the sides, made possible the construction and representation of the polygon. The above results are applied to the study of fundamental regions of the groups cristalográficos not Euclídeos groups (NEC) giving procedures for the construction of regional groups NEC planar surfaces, namely gender topological is 0, and for a parameterization space Teichmà ¼ ller in each case, and resolving accounting and obtaining explicit examples of the group generated in some of the examples solved. The approach to obtain these conditions is algebraic geometry. It works on the models Sl2 (C) and Sl2 (R), which are sub-array 2x2 complex (respectively, actual) trace zero. In addition to working with arrays, the formulas allow for easier handling and are more suited to computer calculations. The report is divided into four chapters and an appendix. The Appendix includes several worksheets (worksheet) made MAPLE, allowing the reader to use any general formulas obtained in the report for polygons Hyperbolic and specific sites of a specific number of sides. They also allow verification of the numerical results and graphs of all the examples contained in this Report.