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In this article we study the space of left- and bi-invariant orderings on atorsion-free nilpotent group $G$. We will show that generally the set of suchorderings 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 $G$ isassumed to be merely residually torsion-free nilpotent. In particular, weobtain faithful actions of mapping class groups of surfaces. We will drawconnections between the structure of orderings on residually torsion-freenilpotent, hyperbolic groups and their Gromov boundaries, and we show that inthose cases a faithful $\Aut(G)$-action on the boundary is equivalent to afaithful $\Aut(G)$ action on the space of left--invariant orderings.

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