Rigidity of braid group actions on ℝ and of low-genus mapping class group actions on S 1

Every nontrivial action of the braid group B n on ℝ by orientation-preserving homeomorphisms yields, up to conjugation by a homeomorphism of ℝ , a representation ρ : B n → error + ( S 1 ) and therefore determines a translation number for every element of B n . In this manuscript we offer a simple characterisation of which actions of B n on ℝ produce translation numbers that agree with those arising from the standard Nielsen–Thurston action on ℝ . Our approach is to prove an analogous statement concerning left orderings of B n via a technique that uses the space of left orderings of B n , the isolated points in this space, and the natural conjugacy action of B n . We use this result to extend recent rigidity results of Mann and Wolff concerning mapping class group actions on S 1 to the case of low-genus surfaces with marked points.