Geometric embeddings of braid groups do not merge conjugacy classes

TL;DR

This paper proves that geometric embeddings of braid groups preserve conjugacy classes, meaning conjugate elements in the larger group originate from conjugate elements in the smaller group, highlighting a structural property of braid group embeddings.

Contribution

It establishes that geometric embeddings of braid groups do not merge conjugacy classes, a property not generally true for mapping class groups.

Findings

Geometric embeddings preserve conjugacy classes in braid groups.

Adding trivial strands does not merge conjugacy classes.

The property does not extend to mapping class groups.

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 B_m into the braid group on n strands B_n, 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 B_m by a geometric embedding are conjugate in B_n, the original elements are conjugate in B_m. We also show that the result does not hold, in general, for geometric embeddings of mapping class groups.