the group acts faithfully on the line

Question

Let $G$ be a group. $G$ acts faithfully on the line $\mathbb{R}$ by orientation preserving homeomorphism, then does it imply $G$ is left ordered, i.e. there is an order $<$ on $G$, and if $a<b$, then $ac<bc$ for $c\in G$? Thanks in advance.

Answer 1

Theorem 1. The group $H=Homeo_+({\mathbb R})$ (of orientation-preserving homeomorphisms of the real line) is left-orderable. In particular, every subgroup $G<H$ is also LO.

Proof. Let $\{x_i: i\in {\mathbb N}\}$ be a countable dense subset of ${\mathbb R}$. For distinct elements $f, g\in H$ let $n$ be the least integer such that $f(x_n)\ne g(x_n)$ (such $n$ always exists of course). Then set
$$ f<g $$ iff $f(x_n)<g(x_n)$. It is then immediate that $<$ is a left order on $H$. qed

A partial converse to this theorem is also true:

Theorem 2. Every countable LO group $G$ is isomorphic to a subgroup of $Homeo_+({\mathbb R})$.