Topology Seminar: Adam Clay

When a group acts faithfully by orientation-preserving homeomorphisms on S^1, one can sometimes use the action on S^1 to prove the existence of a faithful order-preserving action by homeomorphisms on the real line. This can be reworded in algebraic terms by using circular orderings and left-orderings of groups: a circularly orderable group may secretly admit a left-ordering, though the existence of such an ordering may not be apparent.

In this talk I will review some classical results that use cohomology and circular orderings of a group to detect the existence of left-orderings. I'll also present one new technique for determining when a circularly orderable group admits a left-ordering.  As a bonus, when a group is circularly orderable but NOT left-orderable, our new approach determines a subset of the natural numbers that precisely encodes the obstruction to the existence of a left-ordering.  This is joint work with Ty Ghaswala and Jason Bell.