Hostname: page-component-cb9f654ff-mx8w7 Total loading time: 0 Render date: 2025-09-10T22:29:07.751Z Has data issue: false hasContentIssue false
  • English
  • Français

Yet another Poincaré Polyhedron Theorem

Part of: Real and complex geometry Topological geometry Lie groups

Published online by Cambridge University Press:  07 April 2011

Sasha Anan'in  and
Carlos H. Grossi
Sasha Anan'in
Affiliation:
Departamento de Matemática, IMECC, Universidade Estadual de Campinas, 13083-970-Campinas-SP, Brasil (ananin sasha@yahoo.com)
Carlos H. Grossi
Affiliation:
Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany (grossi_ferreira@yahoo.com)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Poincaré'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 Poincaré's Polyhedron Theorem that is applicable to constructing fibre bundles over surfaces and also suits geometries of non-constant curvature. Most conditions of the theorem, being as local as possible, are easy to verify in practice.

Keywords

Poincaré's Polyhedron Theoremdiscrete groupsgeometric structuresfibre bundles

MSC classification

Secondary: 51M10: Hyperbolic and elliptic geometries (general) and generalizations 22E40: Discrete subgroups of Lie groups 51H05: General theory

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 2 , June 2011 , pp. 297 - 308
Copyright
Copyright © Edinburgh Mathematical Society 2011

References

1.Alexandrov, A. D., On tiling a space with polyhedra, in A. D. Alexandrov: selected works I, Classics of Soviet Mathematics, Volume 4 (Gordon and Breach, London, 1996).Google Scholar
2.Anan'in, S., Grossi, C. H. and Gusevskii, N., Complex hyperbolic structures on disc bundles over surfaces, Int. Math. Res. Not. (2010), doi:10.1093/imrn/rnq161.CrossRefGoogle Scholar
3.Beardon, A. F., The geometry of discrete groups, Graduate Texts in Mathematics, Volume 91 (Springer, 1983).CrossRefGoogle Scholar
4.Epstein, D. B. A. and Petronio, C., An exposition of Poincaré's Polyhedron Theorem, Enseign. Math. 40 (1994), 113–170.Google Scholar
5.Kuiper, N. H., Hyperbolic 4-manifolds and tessellations, Publ. Math. IHES 68 (1988), 47–76.CrossRefGoogle Scholar