Moscow Mathematical Journal

Volume 14, Issue 4, October–December 2014 pp. 645–667.

Poincaré's Polyhedron Theorem for Cocompact Groups in Dimension 4

Authors: Sasha Anan’in (1), Carlos H. Grossi (2), and Júlio C. C. da Silva (3)
Author institution:(1) Departamento de Matemática, ICMC, Universidade de São Paulo, Caixa Postal 668, 13560-970–São Carlos–SP, Brasil
(2) Departamento de Matemática, ICMC, Universidade de São Paulo, Caixa Postal 668, 13560-970–São Carlos–SP, Brasil
(3) Departamento de Matemática, IMECC, Universidade Estadual de Campinas, 13083-970–Campinas–SP, Brasil

Summary:

We prove a version of Poincaré’s polyhedron theorem whose requirements are as local as possible. New techniques such as the use of discrete groupoids of isometries are introduced. The theorem may have a wide range of applications and can be generalized to the case of higher dimension and other geometric structures. It is planned as a first step in a program of constructing compact ℂ-surfaces of general type satisfying c₁² = 3c₂.

2010 Mathematics Subject Classification. Primary: 22E40; Secondary: 14J29, 20L05.

Keywords: Poincaré's Polyhedron Theorem, discrete groups, geometric structures on manifolds, compact ℂ-surfaces of general type

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