Acylindrical hyperbolicity, non-simplicity and SQ\hbox{-}universality of groups splitting over ℤ

Abstract

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