Orderings and flexibility of some subgroups of $Home{o}_{+}(\mathbb{R})$

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 $G={\mathbb{F}}_n{\ast }_{\mathbb{Z}}{\mathbb{F}}_m$ where $n+m\ge 3$, 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 $\mathcal{L}\mathcal{O}(G)$, 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 $Home{o}_{+}(\mathbb{R})$ is connected, and also that the natural conjugation action of ${\pi }_1(\Sigma )$ on $\mathcal{L}\mathcal{O}({\pi }_1(\Sigma ))$ has a dense orbit.

27 pages, theorem about amalgams added

Cited by in corpus (4)

• Borel structures on the space of left-orderings
• Rigidity and geometricity for surface group actions on the circle
• Quasi-invariant measures for some amenable groups acting on the line
• Spaces of orders of some one-relator groups