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

TL;DR

This paper studies the flexibility and structure of groups acting on the real line via order-preserving homeomorphisms, revealing new approximation properties, the Cantor set structure of left orders, and connectedness of representation spaces.

Contribution

It demonstrates that certain amalgamated free groups have dense approximation properties in their actions on the line, and establishes the connectedness and density properties of representation and order spaces for surface groups.

Findings

The space of left orders of the group is a Cantor set.

Actions of certain groups can be approximated by others without fixed points.

The space of fixed-point-free representations of surface groups is connected.

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.