Surface groups among cubulated hyperbolic and one-relator groups

Henry Wilton

Surface groups among cubulated hyperbolic and one-relator groups

Group Theory 2026-01-23 v3 Geometric Topology

Authors: Henry Wilton

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.

Keywords

hyperbolic groups mapping class groups group theory

Cite

@article{arxiv.2406.02121,
  title  = {Surface groups among cubulated hyperbolic and one-relator groups},
  author = {Henry Wilton},
  journal= {arXiv preprint arXiv:2406.02121},
  year   = {2026}
}

Comments

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

Surface groups among cubulated hyperbolic and one-relator groups

Abstract

Keywords

Cite

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