New York Journal of Mathematics
Volume 18 (2012) 261-273


Dave Witte Morris

The space of bi-invariant orders on a nilpotent group

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Published: April 16, 2012
Keywords: invariant order, left-orderable group, nilpotent group
Subject: Primary 20F60; Secondary 06F15, 20F18

We prove a few basic facts about the space of bi-invariant (or left-invariant) total order relations on a torsion-free, nonabelian, nilpotent group G. For instance, we show that the space of bi-invariant orders has no isolated points (so it is a Cantor set if G is countable), and give examples to show that the outer automorphism group of G does not always act faithfully on this space. Also, it is not difficult to see that the abstract commensurator group of G has a natural action on the space of left-invariant orders, and we show that this action is faithful. These results are related to recent work of T. Koberda that shows the automorphism group of G acts faithfully on this space.

Author information

Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Alberta, T1K 3M4, Canada