Topology Seminar: Dale Rolfsen

An ordered group (G,) is a group G together with a strict total ordering of its elements which is invariant under left- and right-multiplication. If such an ordering exists for a group, the group is said to be orderable. It is easy to see that if G and H are orderable, then so is their direct product. In 1949, A. A. Vinogradov proved that if G and H are orderable groups, then the free product G*H is also orderable. I’ll show that such an ordering can be constructed in a functorial manner, in the category of ordered groups and order-preserving homomorphisms, using an algebraic trick due to G. Bergman. This was motivated by a certain question in the theory of the braid groups B_n and the Artin representation of B_n in the automorphism group Aut(F_n) of a free group.