  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.
