From Riches to Raags: 3-Manifolds, Right-Angled Artin Groups, and Cubical Geometry (Cbms Regional Conference Series in Mathematics)

This book presents an introduction to the geometric group theory associated with nonpositively curved cube complexes. It advocates the use of cube complexes to understand the fundamental groups of hyperbolic 3-manifolds as well as many other infinite groups studied within geometric group theory. The main goal is to outline the proof that a hyperbolic group G with a quasiconvex hierarchy has a finite index subgroup that embeds in a right-angled Artin group. The supporting ingredients of the proof are sketched: the basics of nonpositively curved cube complexes, wallspaces and dual CAT(0) cube complexes, special cube complexes, the combination theorem for special cube complexes, the combination theorem for cubulated groups, cubical small-cancellation theory, and the malnormal special quotient theorem. Generalizations to relatively hyperbolic groups are discussed. Finally, applications are described towards resolving Baumslag's conjecture on the residual finiteness of one-relator groups with torsion, and to the virtual specialness and virtual fibering of certain hyperbolic 3-manifolds, including those with at least one cusp. The text contains many figures illustrating the ideas. A co-publication of the AMS and CBMS. This book presents an introduction to the geometric group theory associated with nonpositively curved cube complexes. It advocates the use of cube complexes to understand the fundamental groups of hyperbolic 3-manifolds as well as many other infinite groups studied within geometric group theory. The main goal is to outline the proof that a hyperbolic group G with a quasiconvex hierarchy has a finite index subgroup that embeds in a right-angled Artin group. The supporting ingredients of the proof are sketched: the basics of nonpositively curved cube complexes, wallpapers and dual CAT(0) cube complexes, special cube complexes, the combination theorem for special cube complexes, the combination theorem for cubulated groups, cubical small-cancellation theory, and the malnormal special quotient theorem. Generalizations to relatively hyperbolic groups are discussed. Finally, applications are described towards resolving Baumslag's conjecture on the residual finiteness of one-relator groups with torsion, and to the virtual specialness and virtual fithering of certain hyperbolic 3-manifolds, including those with at least one cusp. The text contains many figures illustrating the ideas Overview -- Nonpositively Curved Cube Complexes -- Cubical Disk Diagrams, Hyperplanes, And Convexity -- Special Cube Complexes -- Virtual Specialness Of Malnormal Amalgams -- Wallspaces And Their Dual Cube Complexes -- Finiteness Properties Of The Dual Cube Complex -- Cubulating Malnormal Graphs Of Cubulated Groups -- Cubical Small-cancellation Theory --walls In Cubical Small-cancellation Theory -- Annular Diagrams -- Virtually Special Quotients -- Hyperbolicity And Quasiconvexity Detection -- Hyperbolic Groups With A Quasiconvex Hierarchy -- The Relatively Hyperbolic Setting -- Applications. Daniel T. Wise. Includes Bibliographical References And Index.