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

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.