Nielsen–Thurston orders and the space of braid orderings

We study the topological space of left‐orderings of the braid group, and its subspace of Nielsen–Thurston orderings. Our main result is that no Nielsen–Thurston ordering is isolated in the space of braid orderings. In the course of the proof, we classify the convex subgroups and calculate the Conradian soul for any Nielsen–Thurston ordering of B n . We also prove that, for a large class of Nielsen–Thurston orderings, including all those of infinite type, a stronger result holds: they are approximated by their own conjugates. On the other hand, we suggest an example of a Nielsen–Thurston ordering that may not be approximated by its conjugates.