On the geometry of positive cones in finitely generated groups

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.