Dynamics of isolated left orders

Abstract

A left order of a countable group $G$ is called isolated if it is an isolated point in the compact space $LO(G)$ of all the left orders of $G$. We study properties of a dynamical realization of an isolated left order. Especially we show that it acts on $\mathbb{R}$ cocompactly. As an application, we give a dynamical proof of the Tararin theorem which characterizes those countable groups which admit only finitely many left orders. We also show that the braid group $B_3$ admits countably many isolated left orders which are not the automorphic images of the others.