On the dynamics of endomorphisms of the direct product of two free groups

TL;DR

This paper establishes the decidability of Brinkmann's problems for endomorphisms of the direct product of two free groups, analyzes the dynamics at infinity, and extends results to conjugacy problems in related group extensions.

Contribution

It proves the decidability of Brinkmann's problems for endomorphisms of F_n×F_m and analyzes automorphism dynamics at infinity, extending previous work to new group contexts.

Findings

Decidability of Brinkmann's problems for endomorphisms of F_n×F_m

Decidability of a two-sided conjugacy problem for injective endomorphisms

Automorphisms exhibit asymptotically periodic dynamics at infinity

Abstract

We prove that Brinkmann's problems are decidable for endomorphisms of $F_n\times F_m$: given $(x,y),(z,w)\in F_n\times F_m$ and $\Phi\in \text{End}(F_n\times F_m)$, it is decidable whether there is some $k\in \mathbb{N}$ such that $(x,y)\Phi^k=(z,w)$ (or $(x,y)\Phi^k\sim(z,w)$). We also prove decidability of a two-sided version of Brinkmann's conjugacy problem for injective endomorphisms which, from the work of Logan, yields a solution to the conjugacy problem in ascending HNN-extensions of $F_n\times F_m$. Finally, we study the dynamics of automorphisms of $F_n\times F_m$ at the infinity, proving that that their dynamics at the infinity is asymptotically periodic, as occurs in the free and free-abelian times free cases.