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The Structure of Groups with a Quasiconvex Hierarchy

(AMS-209)

Daniel T. Wise

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Princeton University Press img Link Publisher

Naturwissenschaften, Medizin, Informatik, Technik / Mathematik

Beschreibung

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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Schlagwörter

Hyperbolic 3-manifold, Subsequence, Dimension, Intersection (set theory), Disjoint sets, Topology, Inclusion map, Virtually, Abelian group, Simplicial complex, Equivalence relation, Dual curve, Lexicographical order, Euler characteristic, Automorphism, Fiber bundle, Infimum and supremum, Dimension (vector space), Curvature, Quasi-isometry, Quasiconvex function, Counterexample, Incompressible surface, Monogon, Coset, Summation, HNN extension, Basis (linear algebra), Topological space, Commutator, Möbius transformation, Triangle group, Induced path, Combinatorial map, Torsion group, Generating set of a group, Braid group, Homotopy, Metric space, Characteristic subgroup, Tree (data structure), Injective function, Fundamental group, Theorem, Quotient space (topology), Conjugacy class, Cyclic group, Combination, Double coset, Dual graph, Degeneracy (mathematics), Connected space, Relatively hyperbolic group, Rectangle, Cayley graph, Combinatorial group theory, Artin group, Equivalence class, Disjoint union, Parity (mathematics), Total order, Hyperplane, Convex set, Coxeter group, Special case, Graph of groups, Diagram (category theory), Hyperbolic group, C0, Subgroup