Tubular groups and non positive curvature

TL;DR

This paper investigates the geometric properties of tubular groups, proving they are virtually special when based on a tree graph, and demonstrates the existence of CAT(0) 1-relator groups that are not residually finite.

Contribution

It establishes conditions under which tubular groups are virtually special and constructs examples of CAT(0) 1-relator groups lacking residual finiteness.

Findings

Tubular groups over trees are virtually special.

Existence of CAT(0) 1-relator groups that are not residually finite.

Extension of Wise and Woodhouse's results to broader classes of groups.

Abstract

We show (using results of Wise and of Woodhouse) that a tubular group is always virtually special (meaning that it has a finite index subgroup embedding in a RAAG) if the underlying graph is a tree. We also adapt Gardam and Woodhouse's argument on tubular groups which double cover 1-relator groups to show there exist 1-relator groups which are CAT(0) but not residually finite.