New York Journal of Mathematics
Volume 17 (2011) 783-798


Thomas Koberda

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

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Published: December 5, 2011
Keywords: Orderings on groups, automorphisms of groups, Gromov hyperbolic groups
Subject: Primary: 20F67; Secondary: 20E36

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.


The author was supported by an NSF Graduate Research Fellowship for part of the time that this research was carried out.

Author information

Department of Mathematics, Harvard University, 1 Oxford St., Cambridge, MA 02138