Relative cubulation of relative strict hyperbolization

TL;DR

The paper develops relative strict hyperbolization to produce relatively hyperbolic groups $G=\pi_1(\mathcal{R}(K,L))$ that act cocompactly on CAT(0) cubical complexes, with parabolic stabilizers and higher-codimension stabilizers classified as either maximal parabolic or hyperbolic and virtually compact special. By constructing a dual cubical complex from mirrors on branched covers and proving NPC and simple connectivity, it shows that the $G$-action satisfies a robust cubulation framework even though the action is not proper. Under mild assumptions on peripheral subgroups, the authors prove residual finiteness and separability of peripherals, and in the hyperbolic, virtually compact special case, $G$ itself is hyperbolic and virtually compact special. These results yield manifold applications: new closed aspherical manifolds with residually finite fundamental groups, cobordisms with residually finite fundamental groups, and examples of non-triangulable aspherical manifolds with virtually special fundamental groups, expanding the bridge between hyperbolization, cubulations, and geometric topology.

Abstract

We prove that many relatively hyperbolic groups obtained by relative strict hyperbolization admit a cocompact action on a CAT(0) cubical complex. Under suitable assumptions on the peripheral subgroups, these groups are residually finite and even virtually special. We include some applications to the theory of manifolds, such as the construction of new non-positively curved Riemannian manifolds with residually finite fundamental group, and the existence of non-triangulable aspherical manifolds with virtually special fundamental group.

Figures (5)

• Figure 1: The relative strict hyperbolization procedure.
• Figure 2: Some covering spaces, local pictures around a cone/branch point.
• Figure 3: The dual cubical complex $\mathcal{C}(\widehat{X_\Gamma})$ superimposed on the stra-ti-fi-ca-tion of $\widehat{X_\Gamma}$. Key: $\fullmoon$, $\astrosun$, and $\newmoon$ denote a vertex of height $0$, $1$, and $2$.
• Figure 4: A minimal bridge $p$ supported by $M$, and its projection to $\mathcal{C}(M)$.
• Figure 5: The construction of the hyperbolized mapping torus $\mathcal{T}(M)$.

Theorems & Definitions (62)

• Theorem A
• Theorem B
• Lemma 2.1: Gromov's link condition
• Lemma 2.2
• Remark 3.1
• Remark 3.2
• Remark 3.3: A standing assumption
• Example 3.4
• Remark 3.5
• Remark 3.6: The case of $L$ with simply connected components