The Conjugacy Problem for ascending HNN-extensions of free groups

TL;DR

This work establishes the decidability of the Conjugacy Problem for ascending HNN-extensions of finitely generated free groups by developing a comprehensive dynamical framework for free-group endomorphisms. It introduces automorphic expansions and graph-of-roses models to faithfully represent endomorphisms, and proves a stability phenomenon for iterated pullbacks via the stable iterate $\hat{\Lambda}_k$, enabling finite, algorithmic analysis of long-range dynamics. Central contributions include two generalisations of Brinkmann’s conjugacy algorithm (and a corresponding equality algorithm) that handle injective and non-injective endomorphisms, plus a robust Twisted Conjugacy framework that underpins the main decidability result. The methods extend to the Druţu–Sapir one-relator group, illustrating that ascended free-group HNN-extensions capture a broad and important class of groups with decidable Conjugacy Problems, and highlighting deep connections to dynamics, fixed subgroups, and free-factor structures in free groups.

Abstract

We give an algorithm to solve the Conjugacy Problem for ascending HNN-extensions of free groups. To do this, we give algorithms to solve certain problems on dynamics of free group endomorphisms.

Theorems & Definitions (57)

• Theorem 1.1: Theorem \ref{['thm:CPBODY']}
• Theorem 1.2: Theorem \ref{['thm:UltraTwCPBODY']}
• Theorem 1.3: Theorem \ref{['thm:equalityBODY']}
• Corollary 2.1: Mutanguha, JPM
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Proposition 2.4
• proof