Orderings and flexibility of some subgroups of $Homeo_+(\mathbb{R})$

Abstract: In this work we exhibit flexibility phenomena for some (countable) groups acting by order preserving homeomorphisms of the line. More precisely, we show that if a left orderable group admits an amalgam decomposition of the form $G=\mathbb{F}n*{\mathbb Z} \mathbb{F}m$ where $n+m\geq 3$, then every faithful action of $G$ on the line by order preserving homeomorphisms can be approximated by another action (without global fixed points) that is not semi-conjugated to the initial action. We deduce that $\mathcal{LO}(G)$, the space of left orders of $G$, is a Cantor set. In the special case where $G=\pi_1(\Sigma)$ is the fundamental group of a closed hyperbolic surface, we found finer techniques of perturbation. For instance, we exhibit a single representation whose conjugacy class in dense in the space of representations. This entails that the space of representations without global fixed points of $\pi_1(\Sigma)$ into $Homeo+(\mathbb R)$ is connected, and also that the natural conjugation action of $\pi_1(\Sigma)$ on $\mathcal{LO}(\pi_1(\Sigma))$ has a dense orbit.