Geometric embeddings of braid groups do not merge conjugacy classes

Abstract

An embedding of the m-times punctured disc into the n-times punctured disc, for n > m, yields an embedding of the braid group on m strands Bm into the braid group on n strands Bn, called a geometric embedding. The main example consists of adding n−m trivial strands to the right of each braid on m strands. We show that geometric embeddings do not merge conjugacy classes, meaning that if the images of two elements in Bm by a geometric embedding are conjugate in Bn, the original elements are conjugate in Bm. We also show that the result does not hold, in general, for geometric embeddings of mapping class groups.