Classifying Finite Dimensional Cubulations of Tubular Groups

TL;DR

This paper investigates when tubular groups can act freely on finite dimensional CAT(0) cube complexes, providing criteria, explaining non-separability phenomena, and linking infinite dimensional cubulations to subgroup distortion.

Contribution

It extends Wise's criterion to classify finite dimensional cubulations of tubular groups and connects cubulation properties with subgroup distortion.

Findings

A criterion for finite dimensional cubulation of tubular groups

Explanation of non-separability in 3-manifold groups

Infinite dimensional cubulations imply quadratic subgroup distortion

Abstract

A tubular group is a group that acts on a tree with $\mathbb{Z}^2$ vertex stabilizers and $\mathbb{Z}$ edge stabilizers. This paper develops further a criterion of Wise and determines when a tubular group acts freely on a finite dimensional CAT(0) cube complex. As a consequence we offer a unified explanation of the failure of separability by revisiting the non-separable 3-manifold group of Burns, Karrass and Solitar and relating it to the work of Rubinstein and Wang. We also prove that if an immersed wall yields an infinite dimensional cubulation then the corresponding subgroup is quadratically distorted.