Publicación: Finitely generated non-cocompact NEC groups
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2021
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Atribución-NoComercial-SinDerivadas 4.0 Internacional
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Universidad Nacional de Educación a Distancia (España). Escuela Internacional de Doctorado. Programa de Doctorado en Ciencias
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Esta tesis está dedicada al estudio de grupos discretos de isometrías Г del plano hiperbólico H incluyendo transformaciones que revierten la orientación (reflexiones y reflexiones con desplazamiento) y elementos de contorno (parabólicos e hiperbólicos), de forma que el espacio de órbitas H/Г es no compacto. Dos casos específicos relacionados con los grupos NEC no cocompactos finitimante generados, los subgrupos de isometrías que preservan la orientación o grupos fuchsianos, y los grupos NEC cocompactos han sido ampliamente estudiados en la bibliografía. Este trabajo cubre una laguna que ha existido en la literatura por cierto tiempo introduciendo de forma razonablemente completa los grupos NEC finitamente generados no cocompactos. Se proporciona con demostración la presentación en forma de generadores y relaciones de estos grupos, introduciendo su signatura y usándola para estudiar sus espacios de órbitas y las condiciones necesarias y suficientes de isomorfía entre grupos NEC. Se introduce además un conjunto de invariantes que clasifica las superficies de Klein no compactas salvo homeomorfismos a partir de la signatura del grupo NEC de la que es espacio de órbitas. Obtenemos la característica de Euler del espacio de órbitas y se usa para deducir la signatura del subgrupo fuchsiano canónico de un grupo NEC dada su signatura. Finalmente, se introduce el concepto de grupo NEC elemental y se obtiene la presentación de todos los grupos NEC elementales. Se presentan resultados relacionados con los conjuntos límite de los grupos NEC y se aplican para su clasificación en primer y segundo tipo de forma similar a como se hace con los grupos fuchsianos. Para ello se usan las propiedades del subgrupo fuchsiano canónico del grupo NEC dado.
This thesis is devoted to the study of finitely generated discrete subgroups Г of the whole group of isometries of the hyperbolic plane H including those which reverse the orientation (reflections and glide reflections) as well as boundary transformations (parabolic and boundary hyperbolic elements), such that the orbit space H/Г is not compact. Two special cases closely related to finitely generated non-cocompact NEC groups, the finitely generated discrete subgroups of orientation-preserving isometries (fuchsian groups) and the cocompact NEC groups have been extensively studied in the literature. This work presents a fairly complete introduction of the non-cocompact NEC groups, providing with proof their presentation, introducing their signatures and using them for studying their orbit spaces and the necessary and sufficient conditions of isomorphism between these groups. We present additionally a set of invariants that classify the non-compact Klein surfaces up to homeomorphisms using the signature of the NEC group of which the Klein surface is the orbit space. The Euler characteristic of the orbit space of an NEC group is calculated. Using this we obtain the signature of the non-cocompact canonical fuchsian group linked to the signature of a given NEC group. Finally, the concept of elementary NEC groups is introduced and all the possible elementary groups deduced. Using the properties of their canonical fuchsian groups, some results describing the limit sets of NEC groups are obtained. That leads us to introduce a classification of NEC groups of first and second kind similarly as for fuchsian groups.
This thesis is devoted to the study of finitely generated discrete subgroups Г of the whole group of isometries of the hyperbolic plane H including those which reverse the orientation (reflections and glide reflections) as well as boundary transformations (parabolic and boundary hyperbolic elements), such that the orbit space H/Г is not compact. Two special cases closely related to finitely generated non-cocompact NEC groups, the finitely generated discrete subgroups of orientation-preserving isometries (fuchsian groups) and the cocompact NEC groups have been extensively studied in the literature. This work presents a fairly complete introduction of the non-cocompact NEC groups, providing with proof their presentation, introducing their signatures and using them for studying their orbit spaces and the necessary and sufficient conditions of isomorphism between these groups. We present additionally a set of invariants that classify the non-compact Klein surfaces up to homeomorphisms using the signature of the NEC group of which the Klein surface is the orbit space. The Euler characteristic of the orbit space of an NEC group is calculated. Using this we obtain the signature of the non-cocompact canonical fuchsian group linked to the signature of a given NEC group. Finally, the concept of elementary NEC groups is introduced and all the possible elementary groups deduced. Using the properties of their canonical fuchsian groups, some results describing the limit sets of NEC groups are obtained. That leads us to introduce a classification of NEC groups of first and second kind similarly as for fuchsian groups.
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Palabras clave
hyperbolic plane, non-euclidean chrystallographic groups, finitely generated groups of hyperbolic isometries, non-cocompact NEC groups
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Facultades y escuelas::Facultad de Ciencias
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Programa de doctorado en ciencias