Orbibundles, complex hyperbolic manifolds and geometry over algebras

Author(s)

Hugo Cattarucci Botos

• Document type: Doctoral Thesis
• Press: São Carlos.
• Institution: Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB)
• Defense date: 2022-05-12

Examining board members

Carlos Henrique Grossi Ferreira; Mikhail Viktorovich Belolipetskiy; Igor Mencattini; Ricardo Augusto Emmanuel Mendes

• Advisor: Carlos Henrique Grossi Ferreira

Abstract

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). (AU)

• FAPESP's process: 18/10522-8 - Manifolds modelled on the holomorphic 2-ball
• Grantee: Hugo Cattarucci Botós
• Support Opportunities: Scholarships in Brazil - Doctorate