A proof of Seidel\'s conjectures on the volume of ideal tetrahedra in hyperbolic 3-space

• Ano de defesa: 2017
• Autor(a) principal: Cussy, Omar Chavez
• Tipo de documento: Dissertação
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Biblioteca Digitais de Teses e Dissertações da USP
• Palavras-chave em Português: Conjecturas de Seidel; Espaço hiperbólico real; Real hyperbolic space; Seidel's conjectures; Volume de tetraedros ideais; Volume of ideal tetrahedra
• Link de acesso: http://www.teses.usp.br/teses/disponiveis/55/55135/tde-11092017-161403/

We prove a couple of conjectures raised by J. J. Seidel in On the volume of a hyperbolic simplex, Stud. Sci. Math. Hung. (21, 243249, 1986). These conjectures concern the volume of ideal hyperbolic tetrahedra in hyperbolic 3-space and are related to the following general framework. Since explicit formulae for geometric quantities in hyperbolic space (distance, area, volume, etc.) typically involve sophisticated transcendental functions, it is desirable (and quite useful in practice) to expresses these geometric quantities as monotonic functions of algebraic maps. Seidels Speculation 1 says that the volume of an ideal tetrahedron in hyperbolic 3-space depends only on the determinant and permanent of the doubly stochastic Gram matrix of its vertices; Speculation 4 claims that the mentioned volume is monotone in both the determinant and permanent. We are able to give affirmative answers to Speculations 1 and 4 by parameterizing the classifying space of (labelled) ideal tetrahedra in a suitable way.