The Structure of Groups with a Quasiconvex Hierarchy
Daniel T. Wise
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Naturwissenschaften, Medizin, Informatik, Technik / Mathematik
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.
Mathematical induction, Infimum and supremum, Diagram (category theory), Dimension (vector space), Lexicographical order, Sphere, Injective function, Theorem, Klein bottle, Topological space, Big O notation, Special case, Dual curve, Induced path, Internal and external angles, Cartesian product, Disjoint union, Quasiconvex function, Quotient space (topology), Abelian group, Equivalence relation, Commutator, Hyperbolic group, Diameter, Homomorphism, Equivalence class, Group action, Metric space, Circumference, Bipartite graph, Quasi-isometry, Dimension, Ellipse, Fundamental domain, Hyperplane, Subsequence, Trichotomy (mathematics), Basis (linear algebra), Equation, Homotopy, Simplicial complex, Graph of groups, Combinatorial group theory, Subgroup, Order of magnitude, Parity (mathematics), Partially ordered set, Connected space, Topology, Pairwise, Conjugacy class, Counterexample, Monogon, Curvature, Existential quantification, Braid group, Relatively hyperbolic group, Triangle group, Convex set, Coxeter group, C0, Total order, Cyclic group, Rectangle, Automorphism, Combination, Möbius transformation, Finite group, Degeneracy (mathematics), Intersection (set theory), Nilpotent, HNN extension, Retract, Torsion group, Tree (data structure), Fundamental group, Generating set of a group, Embedding, Subset, Hyperbolic 3-manifold, Cayley graph, Disjoint sets, Incompressible surface, Convex hull, Coset, Combinatorial map, Dual graph, Notation, Characteristic subgroup, Euler characteristic, Connectedness, Simply connected space, Artin group, Geometric group theory, Summation, Double coset, Inclusion map, Disk (mathematics), Fiber bundle, Virtually