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

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.

Acknowledgements

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
koberda@math.harvard.edu