Ordering Pure Braid Groups on Compact, Connected Surfaces

The purpose of this paper is to answer the following question: Are pure braid groups on compact, connected surfaces bi-orderable? We will prove that the answer is positive for orientable surfaces, and negative for the non-orientable ones. In this section we give the basic definitions and classical results. We also explain what is known about orders on braid groups, and finally we state our results. In Section 2 we study the particular case of closed, orientable surfaces. The closed, non-orientable surfaces are treated in Section 3 and, in Section 4, we extend our results to all compact, connected surfaces. Let us just mention that, if a surface is non-connected, its braid groups are a direct product of braid groups on each connected component (it needs to be taken into acount how many base points are in each connected component). Knowing that a direct product of groups is bi-orderable if and only if each one is bi-orderable, we can extend our results to all compact surfaces.