Faithful actions of automorphisms on the space of orderings of a group

Abstract

In this article we study the space of left- and bi-invariant orderings on a torsion-free nilpotent group $G$. We will show that generally the set of such orderings is equipped with a faithful action of the automorphism group of $G$. We prove a result which allows us to establish the same conclusion when $G$ is assumed to be merely residually torsion-free nilpotent. In particular, we obtain faithful actions of mapping class groups of surfaces. We will draw connections between the structure of orderings on residually torsion-free nilpotent, hyperbolic groups and their Gromov boundaries, and we show that in those cases a faithful $\Aut(G)$-action on the boundary is equivalent to a faithful $\Aut(G)$ action on the space of left--invariant orderings.