Incubulable hyperbolic 3-pseudomanifold groups

TL;DR

This paper constructs specific hyperbolic 3-pseudomanifold groups with unique geometric properties, showing they are hyperbolic but not cubulable, and analyzing their boundary subgroup intersections.

Contribution

It introduces new examples of hyperbolic 3-pseudomanifold groups with property (T) and explores their cubulation properties, revealing novel geometric group theory phenomena.

Findings

Fundamental groups are hyperbolic but not cubulable.

Existence of locally convex subspaces with property (T).

Boundary subgroup intersections are infinite in any relative cubulation.

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.