Yet Another Poincare's Polyhedron Theorem

TL;DR

This paper presents a new version of Poincaré's Polyhedron Theorem that applies to constructing fiber bundles over surfaces and accommodates geometries with nonconstant curvature, with practical local conditions.

Contribution

It introduces a generalized Poincaré's Polyhedron Theorem applicable to fiber bundles and variable curvature geometries, with easily verifiable local conditions.

Findings

Provides a version of the theorem for fiber bundles over surfaces.

Extends applicability to nonconstant curvature geometries.

Conditions are designed to be easy to verify in practice.

Abstract

Poincar\'e's Polyhedron Theorem is a widely known valuable tool in constructing manifolds endowed with a prescribed geometric structure. It is one of the few criteria providing discreteness of groups of isometries. This work contains a version of Poincar\'e's Polyhedron Theorem that is applicable to constructing fibre bundles over surfaces and also suits geometries of nonconstant curvature. Most conditions of the theorem, being as local as possible, are easy to verify in practice.