Special cubulation of strict hyperbolization

TL;DR

This paper proves that hyperbolic groups from strict hyperbolization are virtually compact special, leading to new examples of negatively curved manifolds with linear, residually finite, and virtually fibered fundamental groups.

Contribution

It establishes that these hyperbolic groups are virtually compact special by constructing an action on a dual CAT(0) cubical complex, a novel approach in the context of strict hyperbolization.

Findings

Hyperbolized groups are virtually compact special.

Fundamental groups of resulting manifolds are linear and residually finite.

Provides new examples of negatively curved manifolds with virtually algebraically fibered groups.

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.