Detection of fully irreducible automorphisms in generalized Baumslag-Solitar groups

TL;DR

This paper introduces a new way to identify fully irreducible automorphisms in generalized Baumslag-Solitar groups, providing characterizations, properties, and an algorithm for detection.

Contribution

It extends the concept of fully irreducible automorphisms from free groups to generalized Baumslag-Solitar groups, including a characterization and an algorithm for their detection.

Findings

Characterization of fully irreducible automorphisms

Existence of pseudo-periodic conjugacy classes

Algorithm for detecting pseudo-atoroidal and fully irreducible automorphisms

Abstract

We define fully irreducible automorphisms of generalized Baumslag-Solitar groups in analogy with fully irreducible automorphisms of free groups. We first obtain a characterization of fully irreducible automorphisms analogous to a condition given by Kapovich. Next we discuss the existence of pseudo-periodic conjugacy classes based on the study of Nielsen paths in a train track representative. Finally we give an algorithm which finds out whether an automorphism with a train track representative is pseudo-atoroidal and fully irreducible.