  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.
