Braid groups are not profinitely rigid

We show that the $n$-strand braid group ${\mathbf{B}}_n$ is not profinitely rigid when $n\ge 4$. We do this by exhibiting a family of pairwise non-isomorphic, finitely presented residually finite groups with the same profinite completion as ${\mathbf{B}}_n$. This answers a well-known problem negatively, and gives the first known examples of non-profinitely rigid mapping class groups.