  Let $X$ be a non-positively curved cube complex with hyperbolic fundamental
group. We prove that $\pi_1(X)$ has a non-free subgroup of infinite index
unless $\pi_1(X)$ is either free or a surface group, answering a question of
Wise. A similar result for one-relator groups follows, answering a question
posed by several authors. The proof relies on a careful analysis of free and
cyclic splittings of cubulated groups.
