The Post Correspondence Problem and equalisers for certain free group and monoid morphisms

A marked free monoid morphism is a morphism for which the image of each generator starts with a different letter, and immersions are the analogous maps in free groups. We show that the (simultaneous) PCP is decidable for immersions of free groups, and provide an algorithm to compute bases for the sets, called equalisers, on which the immersions take the same values. We also answer a question of Stallings about the rank of the equaliser. Analogous results are proven for marked morphisms of free monoids.