The Hurwitz Equivalence Problem is Undecidable

In this paper, we prove that the Hurwitz equivalence problem for 1-factorizations in $F_2\oplus F_2$ is undecidable, and as a consequence, the Hurwitz equivalence problem for ${\mathrm{\Delta }}^2$-factorizations in the braid groups $B_n,n\ge 5$ is also undecidable.