  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.
