jats:titleAbstract</jats:title> jats:pWe show, using acylindrical hyperbolicity, that a finitely generated group splitting over <jats:inline-formula id="j_jgth-2016-0045_ineq_9999_w2aab3b8e5114b1b7b1aab1c14b1b1Aa"> jats:alternatives <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ℤ</m:mi> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" content-type="j_jgth-2016-0045_ineq_9999" xlink:href="graphic/j_jgth-2016-0045_eq_mi106.png" /> jats:tex-math$Z$</jats:tex-math> </jats:alternatives> </jats:inline-formula> 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 <jats:inline-formula id="j_jgth-2016-0045_ineq_9998_w2aab3b8e5114b1b7b1aab1c14b1b3Aa"> jats:alternatives <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ℤ</m:mi> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" content-type="j_jgth-2016-0045_ineq_9998" xlink:href="graphic/j_jgth-2016-0045_eq_mi106.png" /> jats:tex-math$Z$</jats:tex-math> </jats:alternatives> </jats:inline-formula> must either be SQ-universal or it is one of exactly seven virtually abelian exceptions.</jats:p>