Orderable 3-manifold groups
by
Dale Rolfsen
University of British Columbia
Coauthors: Steve Boyer (UQAM), Bert Wiest (UBC-PIMS pdf)

A group is said to be left-orderable if its elements can be given a strict total ordering < which is invariant under left-multiplication, and bi-orderable if such an ordering exists which is simultaneously left- and right invariant. For fundamental groups of spaces, I will discuss the topological and algebraic consequences of orderability, and techniques for constructing orderings. In particular, we will see that surprisingly many 3-manifold groups are orderable. For example all knot groups are left-orderable, though not all are bi-orderable; torus knot groups cannot be bi-orderded, but the figure-eight knot group can. At the current state of the art, we understand the orderability properties of geometric 3-manifolds in seven of the eight 3-dimensional geometries - hyperbolic remaining mysterious.

Date received: March 28, 2001