Automorphisms acting on the left-orderings of a bi-orderable group

TL;DR

This paper proves that automorphism groups act faithfully on the space of left-orderings for nonabelian bi-orderable groups and certain other groups, including braid groups, with implications for hyperbolic groups.

Contribution

It generalizes Koberda's result by establishing faithfulness of automorphism actions on left-orderings for a broader class of groups, including bi-orderable and some left-orderable groups.

Findings

Automorphism group action is faithful for all nonabelian bi-orderable groups.

Automorphism group action is faithful on the boundary for bi-orderable hyperbolic groups.

Analyzes the action of the commensurator on virtual left-orderings.

Abstract

We generalize a result of T. Koberda by showing that the natural action of the automorphism group on the space of left-orderings is faithful for all nonabelian bi-orderable groups G, as well as for a certain class of left-orderable groups that includes the braid groups. As a corollary we show that the action of the automorphism group of G on the boundary of G is faithful whenever G is bi-orderable and hyperbolic. We also analyze the action of the commensurator of G on its space of virtual left-orderings.