Poincar\'e's polyhedron theorem for cocompact groups in dimension 4

TL;DR

This paper extends Poincaré's polyhedron theorem to cocompact groups in four dimensions, introducing new techniques like discrete groupoids, with potential applications in higher-dimensional geometry and complex surface construction.

Contribution

It presents a localized version of Poincaré's polyhedron theorem for 4D cocompact groups and introduces innovative methods such as discrete groupoids of isometries.

Findings

Proves a local version of Poincaré's polyhedron theorem for 4D cocompact groups

Introduces new techniques including discrete groupoids of isometries

Suggests applications to higher-dimensional geometry and complex surface theory

Abstract

We prove a version of Poincar\'e'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 $c_1^2=3c_2$.