Acylindrical Hyperbolicity, non-simplicity and SQ-universality of groups splitting over $\mathbb{Z}$

Abstract

We show, using acylindrical hyperbolicity, that a finitely generated group splitting over $\mathbb{Z}$ 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 $\mathbb{Z}$ must either be SQ-universal or it is one of exactly seven virtually abelian exceptions.