Special cubulation of strict hyperbolization

Abstract

We prove that the Gromov hyperbolic groups obtained by the strict hyperbolization procedure of Charney and Davis are virtually compact special, hence linear and residually finite. Our strategy consists in constructing an action of a hyperbolized group on a certain dual CAT(0) cubical complex. As a result, all the common applications of strict hyperbolization are shown to provide manifolds with virtually compact special fundamental group. In particular, we obtain examples of closed negatively curved Riemannian manifolds whose fundamental groups are linear and virtually algebraically fiber.

70 pages, 32 figures, comments are welcome. v2: Corollary 1.6 in v1 needed a relative version of our main theorem, so it has been moved to arXiv:2304.14946. v3: Theorem 1.2 has been improved and now works for any compact simplicial complex. Various minor edits to improve the exposition. Figure 20 is new. v4: minor edits. Final version to appear in Inventiones Mathematicae

References in corpus (18)

• Criteria for virtual fibering
• Simplicite de groupes d'automorphismes d'espaces a courbure negative
• Groups acting on CAT(0) cube complexes
• Finiteness properties of cubulated groups
• Residually finite rationally solvable groups and virtual fibring
• Anti-trees and right-angled Artin subgroups of braid groups
• Hyperbolic groups acting improperly
• Proper affine actions for right-angled Coxeter groups
• Aspherical manifolds, relative hyperbolicity, simplicial volume and assembly maps
• Finiteness of Maximal Geodesic Submanifolds in Hyperbolic Hybrids
• Noncoherence of some lattices in Isom(Hn)
• Non-coherence of arithmetic hyperbolic lattices
• Trees of manifolds as boundaries of spaces and groups
• Aspherical manifolds that cannot be triangulated
• Right-angled Artin subgroups of the $C^{\infty }$ diffeomorphism group of the real line
• Special cubulation of strict hyperbolization
• Manifolds without real projective or flat conformal structures
• Foldable cubical complexes of nonpositive curvature

Cited by in corpus (3)

• Special cubulation of strict hyperbolization
• Cubulated hyperbolic groups admit Anosov representations
• Relative cubulation of relative strict hyperbolization