Orbibundles, complex hyperbolic manifolds and geometry over algebras

• Autor(a) principal: Botos, Hugo Cattarucci
• Data de Publicação: 2022
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: https://www.teses.usp.br/teses/disponiveis/55/55135/tde-26072022-085204/
• topic: Álgebras reais; Complex hyperbolic geometry; Difeologia; Diffeology; Discrete invariants; Geometria hiperbólica complexa; Invariantes discretos; Orbifolds; Orbifolds; Real algebras

This thesis consists of the original works Hugo C. Botós, Orbifolds and orbibundles in complex hyperbolic geometry, arXiv:2011.09372; Hugo C. Botós, Carlos H. Grossi. Quotients of the holomorphic 2-ball and the turnover, arXiv:2109.08753; Hugo C. Botós, Geometry over algebras, arXiv:2203.05101; as well as an analysis of the main results of each one of them. The first work introduced basic tools to deal with orbifolds and orbibundles from a diffeological viewpoint. The focus is on developing tools applicable to the construction of complex hyperbolic manifolds. In the second work, several new examples of disc bundles (over closed surfaces) admitting complex hyperbolic structures are constructed. They originate from disc orbibundles over spheres with three cone points and, as such, admit a non-rigid (deformable) complex hyperbolic structure. All the examples obtained support the Gromov-Lawson-Thurston conjecture. The latter establishes the theory of classic geometries over algebras beyond real numbers, complex numbers, and quaternions. We use these geometries to describe the spaces of oriented geodesics in the hyperbolic plane, the Euclidean plane, and the round 2-sphere. Finally, we present a natural geometric transition between such spaces and build a projective model for the geometry of the hyperbolic bidisc (the Riemannian product of two hyperbolic planes).