On the space of left orders of braid group

Abstract

In this thesis, we will deal with the space of left orders on braid groups. A large class of left orders called Nielsen-Thurston orders is known but the entire space of left orders on braid groups is not yet understood. It is known that every Nielsen-Thurston orders is a limit point in the space. On the other hand, Dubrovina and Dubrovin constructed an isolated left order in [3] by modifying the Dehornoy order that is a Nielsen-Thurston order. We will show that the Dubrovina-Dubrovin order is essentially the only isolated order that can be obtained by modifying a Nielsen-Thurston order.