This article is cited in 1 paper

Poincaré's polyhedron theorem for cocompact groups in dimension $4$

Sasha Anan'in^a, Carlos H. Grossi^a, Júlio C. C. da Silva<sup>b

a</sup> Departamento de Matemática, ICMC, Universidade de São Paulo, Caixa Postal 668, 13560-970—São Carlos—SP, Brasil
^b Departamento de Matemática, IMECC, Universidade Estadual de Campinas, 13083-970—Campinas—SP, Brasil

Abstract: 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 $\mathbb C$-surfaces of general type satisfying $c_1^2=3c_2$.

Key words and phrases: Poincaré's polyhedron theorem, discrete groups, geometric structures on manifolds, compact $\mathbb C$-surfaces of general type.

MSC: Primary 22E40; Secondary 14J29, 20L05

Received: October 29, 2013; in revised form December 14, 2013

Language: English

DOI: 10.17323/1609-4514-2014-14-4-645-667

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