Surface groups among cubulated hyperbolic and one-relator groups

Abstract

Let $X$ be a non-positively curved cube complex with hyperbolic fundamental group. We prove that $\ddot{I}{€}_1(X)$ has a non-free subgroup of infinite index unless $\ddot{I}{€}_1(X)$ is either free or a surface group, answering questions of Gromov and Whyte (in a special case) and 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.

52 pages, 1 figure. v2 incorporates referee comments, and a new §5 answers Question 5.1 from v1. v3 incorporates further referee comments and is the final version accepted for publication