Classifying Virtually Special Tubular Groups

TL;DR

This paper characterizes when tubular groups are virtually special by their actions on CAT(0) cube complexes, establishing conditions for virtual specialness and free actions in finite dimensions.

Contribution

It provides a complete characterization of virtually special tubular groups based on their free actions on CAT(0) cube complexes and relates actions in different dimensions.

Findings

A tubular group is virtually special iff it acts freely on a locally finite CAT(0) cube complex.

If a tubular group acts freely on a finite dimensional CAT(0) cube complex, it virtually acts freely on a 3D CAT(0) cube complex.

The paper establishes a dimension reduction result for free actions of tubular groups.

Abstract

A group is tubular if it acts on a tree with $\mathbb{Z}^2$ vertex stabilizers and $\mathbb{Z}$ edge stabilizers. We prove that a tubular group is virtually special if and only if it acts freely on a locally finite CAT(0) cube complex. Furthermore, we prove that if a tubular group acts freely on a finite dimensional CAT(0) cube complex, then it virtually acts freely on a three dimensional CAT(0) cube complex.