Variations on the Post Correspondence Problem for free groups

TL;DR

This paper explores variations of the Post Correspondence Problem within free groups, establishing connections and providing methods to compute equaliser bases and solve generalized PCP in this context.

Contribution

It introduces new results linking free group homomorphisms to the classical PCP, including algorithms for computing equaliser bases under certain conditions.

Findings

Computing the rank of the equaliser can be used to find a basis.

The generalized PCP for free groups can be solved under specific injectivity assumptions.

Connections between classical and free group PCP variations are established.

Abstract

The Post Correspondence Problem is a classical decision problem about equalisers of free monoid homomorphisms. We prove connections between several variations of this classical problem, but in the setting of free groups and free group homomorphisms. Among other results, and working under certain injectivity assumptions, we prove that computing the rank of the equaliser of a pair of free group homomorphisms can be applied to computing a basis of this equaliser, and also to solve the "generalised" Post Correspondence Problem for free groups.