Incubulable hyperbolic 3-pseudomanifold groups

TL;DR

The paper resolves a version of Cannon-type questions by constructing a sequence of compact hyperbolic $3$-manifolds with totally geodesic boundary whose boundary cone-offs yield closed negatively curved $3$-pseudomanifolds containing infinite quasiconvex subgroups with property $(T)$; consequently, the fundamental groups are word hyperbolic but not cubulable, and no relatively geometric cubulation can avoid having a hyperplane stabilizer with infinite intersection with a boundary subgroup. The approach blends a seed with property $(T)$ built from a girth-enhancing tower of covers, a tailored hyperbolic orbifold with mirror structure, and KM’s cone-off framework to transfer $T$-type subgroups into negative curvature cone-offs of covers. Key contributions include establishing non-cubulability and non-Haagerup/ non-RFRS behavior for the resulting groups, and proving that any relative cubulation forces a strong constraint on hyperplane-stabilizer interactions with boundary subgroups. The work leverages orbifold reflection tricks, Coxeter-word analysis, and controlled coverings to embed $(T)$-subgroups into negatively curved pseudomanifolds while driving topological complexity (genus and Euler characteristics) to infinity, highlighting fundamental distinctions between hyperbolic 3-pseudomanifold groups and genuine 3-manifold groups.

Abstract

We construct compact hyperbolic 3-manifolds with totally geodesic boundary, such that the closed 3-pseudomanifolds obtained by coning off the boundary components are negatively curved and contain locally convex subspaces whose fundamental groups have property (T). In particular, the fundamental groups of these 3-pseudomanifolds are word hyperbolic but not cubulable. We deduce that in any relative cubulation of one of these hyperbolic 3-manifold groups some hyperplane stabilizer has infinite intersection with the fundamental group of some boundary component.

Figures (7)

• Figure 1: The main covering steps in the construction. (The picture only shows a portion of $H$ and $M$.) The boundaries of $M$ (as a manifold) and of $H$, $H_0$, and $B$ (as orbifolds) are represented in green. The boundary of $M$ consists of closed surfaces, the boundaries of $H$ and $H_0$ consist of compact surfaces with boundary, and the boundary of $B$ consists of the face $F_B$.
• Figure 2: Each pair of pants in $Y_0$ in the topological boundary of $H_0$ is cellulated by pentagons (gray) and octagons (yellow) as shown.
• Figure 3: A mirror structure on $H_0$, with $k=4$. The mirror pattern of $Y_0$ (see Figure \ref{['fig:patterned_pants']}) is shown only on the front of the outside pair of pants, but it should be replicated on the other $k-1$ pairs of pants. The boundary of $H_0$ as an orbifold is colored green.
• Figure 4: The pentagonal prism $B$. The unlabeled edges have angle $\pi/2$. Edges labeled by an integer $n$ have angle $\pi/n$. The green face $F_B$ is the image of $S_0$. Its bold edge $e$ is the image of the topological boundary of $S_0$. The yellow face $F_\Sigma$ is the image of the octagonal faces in $Y_0$.
• Figure 5: From left to right: The handlebody $H_0$, with $k=4$; the quotient of $H_0$ by the $D_3$-symmetry; a different representation of the same quotient, in which the $D_k$-symmetry is more clear; the further quotient by the $D_k$-symmetry, i.e., the pentagonal prism $B$, also represented in Figure \ref{['fig:prism']}.

Theorems & Definitions (33)

• Theorem 1.3
• Corollary 1.4
• Remark 1.5: Virtual cubulability
• Remark 1.6: Gromov boundary
• Remark 1.7: Property (T) vs Haagerup Property
• Remark 1.8: Relatively geometric cubulation
• Definition 2.1
• Theorem 2.2
• Definition 2.3
• Theorem 2.4