The Post Correspondence Problem for free groups is undecidable

Abstract

We prove that the Post Correspondence Problem for finitely generated free groups is undecidable, even when one of the two homomorphisms is injective and has finite-index image. This resolves a longstanding open problem in algorithmic group theory. The proof proceeds through a connection with finite-state transducers. Given a cyclic tag system $\mathcal C$, we effectively construct a finite partial deterministic inverse transducer $\mathcal T_{\mathcal C}$ whose fixed-point set is nontrivial if and only if $\mathcal C$ halts. We then associate to any such transducer two homomorphisms $g,h\colon F_Y\longrightarrow F_A,$ with $h$ injective, such that their equalizer is nontrivial precisely when the transducer has a nontrivial fixed loop. As an immediate consequence, the rank of these equalizers cannot be computed in general, answering a question posed by Stallings in 1984. We further prove that there is no algorithm which decides whether the fixed subgroup of a virtual endomorphism of a finitely generated free group is trivial. Finally, we apply the main result to show that the stabilizer problem is undecidable for free subgroups of $\operatorname{SL}_4(\mathbb Z)$, and that the upper-right-corner problem is undecidable for free subgroups of $\operatorname{SL}_5(\mathbb Z)$, even when the given generators are promised to form a free basis, improving on recent results of Breuillard and Kocharyan.