A Cubical Flat Torus Theorem and the Bounded Packing Property

TL;DR

This paper proves the bounded packing property for abelian subgroups in groups acting on CAT(0) cube complexes, introduces a cubical flat torus theorem, and explores implications for group properties and examples.

Contribution

It establishes a cubical flat torus theorem and applies it to analyze subgroup properties and construct examples of groups with specific cubulation characteristics.

Findings

Bounded packing property holds for abelian subgroups in CAT(0) cube complex groups.

Central HNN extensions of maximal free-abelian subgroups are virtually special.

Examples of groups not admitting cocompact cubulations are provided.

Abstract

We prove the bounded packing property for any abelian subgroup of a group acting properly and cocompactly on a CAT(0) cube complex. A main ingredient of the proof is a cubical flat torus theorem. This ingredient is also used to show that central HNN extensions of maximal free-abelian subgroups of compact special groups are virtually special, and to produce various examples of groups that are not cocompactly cubulated.