🤖 AI Summary
This study addresses the decidability of the Post Correspondence Problem (PCP) over free groups when one of the two homomorphisms is injective. By reducing the halting problem for cyclic tag systems to the triviality problem for equalizers of free group homomorphisms and employing constructions based on finite partial deterministic inverse automata, the authors prove that PCP remains undecidable even under this restriction. This result resolves a long-standing open question posed by Stallings in 1984, establishing for the first time that injectivity of one homomorphism does not render the problem decidable. As a corollary, it follows that there is no general algorithm to compute the rank of such equalizers, nor to effectively construct bases or associated automata for fixed subgroups of virtual endomorphisms.
📝 Abstract
We prove that the Post Correspondence Problem for finitely generated free groups is undecidable, even when one of the two homomorphisms is injective. This resolves a longstanding open problem in algorithmic group theory. The result exhibits a sharp contrast with the theory of fixed subgroups: although the equalizer of two free-group homomorphisms is finitely generated whenever one of the maps is injective, there is no algorithm that decides whether this equalizer is trivial.
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. Consequently, the rank of these equalizers cannot be computed in general, answering a question posed by Stallings in 1984.
As a further consequence, we prove that no algorithm computes a basis for the fixed subgroup of a virtual endomorphism of a finitely generated free group, nor constructs a finite automaton recognizing the reduced fixed-point language of a finite complete inverse transducer.