On the intersection of fixed subgroups of $F_n\times F_m$

TL;DR

This paper investigates the decidability of fixed subgroup intersections in free group products, providing an algorithm for certain cases and linking the general problem to the Post Correspondence Problem.

Contribution

It introduces an algorithm to decide fixed subgroup intersections for specific automorphisms and endomorphisms, and relates the general problem to a well-known undecidable problem.

Findings

Decidable intersection problem for certain monomorphisms and endomorphisms.

Undecidability persists for arbitrary endomorphisms.

Connection established between fixed subgroup intersection and Post Correspondence Problem.

Abstract

We prove that, although it is undecidable if a subgroup fixed by an automorphism intersects nontrivially an arbitrary subgroup of $F_n\times F_m$, there is an algorithm that, taking as input a monomorphism and an endomorphism of $F_n\times F_m$, decides whether their fixed subgroups intersect nontrivially. The general case of this problem, where two arbitrary endomorphisms are given as input remains unknown. We show that, when two endomorphisms of a certain type are given as input, this problem is equivalent to the Post Correspondence Problem for free groups.