Surface groups among cubulated hyperbolic and one-relator groups

TL;DR

This paper proves that hyperbolic groups from non-positively curved cube complexes typically contain surface groups unless they are free or surface groups themselves, and extends similar results to one-relator groups.

Contribution

It establishes new conditions under which hyperbolic cubulated groups contain surface subgroups, answering longstanding questions by Gromov, Whyte, and Wise.

Findings

Hyperbolic fundamental groups of non-positively curved cube complexes contain surface groups unless they are free or surface groups.

Similar results are proven for one-relator groups, confirming conjectures.

The proofs involve analyzing free and cyclic splittings of cubulated groups.

Abstract

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 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.