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About This Book
This monograph on the applications of cube complexes constitutes a breakthrough in the fields of geometric group theory and 3-manifold topology. Many fundamental new ideas and methodologies are presented here for the first time, including a cubical small-cancellation theory that generalizes ideas from the 1960s, a version of Dehn Filling that functions in the category of special cube complexes, and a variety of results about right-angled Artin groups. The book culminates by establishing a remarkable theorem about the nature of hyperbolic groups that are constructible as amalgams.The applications described here include the virtual fibering of cusped hyperbolic 3-manifolds and the resolution of Baumslag's conjecture on the residual finiteness of one-relator groups with torsion. Most importantly, this work establishes a cubical program for resolving Thurston's conjectures on hyperbolic 3-manifolds, and validates this program in significant cases. Illustrated with more than 150 color figures, this book will interest graduate students and researchers working in geometry, algebra, and topology.
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Table of contents
- Cover
- Contents
- Acknowledgments
- 1. Introduction
- 2. CAT(0) Cube Complexes
- 3. Cubical Small-Cancellation Theory
- 4. Torsion and Hyperbolicity
- 5. New Walls and the B(6) Condition
- 6. Special Cube Complexes
- 7. Cubulations
- 8. Malnormality and Fiber-Products
- 9. Splicing Walls
- 10. Cutting X*
- 11. Hierarchies
- 12. Virtually Special Quotient Theorem
- 13. Amalgams of Virtually Special Groups
- 14. Large Fillings Are Hyperbolic and Preservation of Quasiconvexity
- 15. Relatively Hyperbolic Case
- 16. Largeness and Omnipotence
- 17. Hyperbolic 3-Manifolds with a Geometrically Finite Incompressible Surface
- 18. Limit Groups and Abelian Hierarchies
- 19. Application Towards One-Relator Groups
- 20. Problems
- References
- Index