Acylindrical hyperbolicity, non simplicity and SQ-universality of groups splitting over Z

TL;DR

This paper demonstrates that finitely generated groups splitting over Z are not simple and are often SQ-universal, with specific exceptions, using acylindrical hyperbolicity techniques.

Contribution

It introduces new results linking acylindrical hyperbolicity to group splitting over Z, establishing non-simplicity and SQ-universality conditions.

Findings

Finitely generated groups splitting over Z cannot be simple.

Most such groups are SQ-universal, with seven virtually abelian exceptions.

The results apply to balanced groups with specific conjugation properties.

Abstract

We show, using acylindrical hyperbolicity, that a finitely generated group splitting over $\Z$ cannot be simple. We also obtain SQ-universality in most cases, for instance a balanced group (one where if two powers of an infinite order element are conjugate then they are equal or inverse) which is finitely generated and splits over $\Z$ must either be SQ-universal or it is one of exactly seven virtually abelian exceptions.