The Structure of Groups with a Quasiconvex Hierarchy

The Structure of Groups with a Quasiconvex Hierarchy


Daniel T. Wise


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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.


Daniel T. Wise:
Daniel T. Wise is James McGill Professor in the Department of Mathematics and Statistics at McGill University. His previous book is From Riches to Raags: 3-Manifolds, Right-Angled Artin Groups, and Cubical Geometry.