On the geometry of positive cones in finitely generated groups

TL;DR

This paper explores the geometric properties of positive cones in finitely generated groups, introducing the Hucha and Prieto properties, and demonstrating their presence in various important classes of groups.

Contribution

It introduces the Hucha and Prieto properties for left-orderable groups and analyzes their stability and presence in free products, limit groups, and subgroups of free -groups.

Findings

All left-orderable free products have the Hucha property.

Hucha property is stable under certain free products with amalgamation over Prieto subgroups.

Non-abelian limit groups and certain subgroups of free -groups have the Hucha property.

Abstract

We study the geometry of positive cones of left-invariant total orders (left-order, for short) in finitely generated groups. We introduce the \textit{Hucha property} and the \texit{Prieto property} for left-orderable groups. The first one means that in any left-order the corresponding positive cone is not coarsely connected, and the second one that in any left-order the corresponding positive cone is coarsely connected. We show that all left-orderable free products have the Hucha property, and that the Hucha property is stable under certain free products with amalgamatation over Prieto subgroups. As an application we show that non-abelian limit groups in the sense of Z. Sela (e.g. free groups, fundamental group of hyperbolic surfaces, doubles of free groups and others) and non-abelian finitely generated subgroups of free $\mathbb{Q}$-groups in the sense of G. Baumslag have the Hucha property. In particular, this implies that these groups have empty BNS-invariant $\Sigma^1$ and that they don't have finitely generated positive cones.