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Surface groups among cubulated hyperbolic and one-relator groups

Henry Wilton

TL;DR

Let $G=\pi_1(X)$ for a compact NPC cube complex $X$ with hyperbolic $G$. The paper develops Whitehead complexes $\mathrm{Wh}_X(Y)$ to model complements and analyzes free and cyclic splittings via 0-, 1-, and 2-cuts, yielding uniform bounds on $k$-cut widths and a criterion that obstructs free splittings. The main result shows that in cubulated hyperbolic groups, unless $G$ is free or a surface group, $G$ has a one-ended quasiconvex subgroup of infinite index, with corollaries extending to virtually special groups and one-relator groups through primitivity-rank technology. These advances address questions of Gromov–Whyte and related researchers in broad classes, and provide an effective geometric framework for Grushko–JSJ-type decompositions via a hierarchy of Whitehead-type obstructions. The work thus advances the surface-subgroup program in cubulated settings, connects local link data to global splittings, and clarifies when infinite-index subgroups must be non-free, offering concrete tools for future higher-dimensional generalizations and algorithmic decompositions.

Abstract

Let $X$ be a non-positively curved cube complex with hyperbolic fundamental group. We prove that $π_1(X)$ has a non-free subgroup of infinite index unless $π_1(X)$ 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.

Surface groups among cubulated hyperbolic and one-relator groups

TL;DR

Let for a compact NPC cube complex with hyperbolic . The paper develops Whitehead complexes to model complements and analyzes free and cyclic splittings via 0-, 1-, and 2-cuts, yielding uniform bounds on -cut widths and a criterion that obstructs free splittings. The main result shows that in cubulated hyperbolic groups, unless is free or a surface group, has a one-ended quasiconvex subgroup of infinite index, with corollaries extending to virtually special groups and one-relator groups through primitivity-rank technology. These advances address questions of Gromov–Whyte and related researchers in broad classes, and provide an effective geometric framework for Grushko–JSJ-type decompositions via a hierarchy of Whitehead-type obstructions. The work thus advances the surface-subgroup program in cubulated settings, connects local link data to global splittings, and clarifies when infinite-index subgroups must be non-free, offering concrete tools for future higher-dimensional generalizations and algorithmic decompositions.

Abstract

Let be a non-positively curved cube complex with hyperbolic fundamental group. We prove that has a non-free subgroup of infinite index unless 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.
Paper Structure (20 sections, 50 theorems, 75 equations, 1 figure)

This paper contains 20 sections, 50 theorems, 75 equations, 1 figure.

Table of Contents

  1. Preliminaries
  2. Non-positively curved cube complxes
  3. CAT(0) cube complexes
  4. Simplicial complexes
  5. Free splittings
  6. Bounding hyperplanes and Whitehead complexes
  7. Ends and splittings
  8. Connected sums of simplicial complexes
  9. Cut-set lemmas in connected sums
  10. Cuts and the generalised Whitehead's lemma
  11. Shepherd's theorem and 1-cuts
  12. Cyclic splittings
  13. Periodic 2-cuts
  14. A bound on the widths of 2-cuts
  15. From 2-cuts to k-cuts
  16. ...and 5 more sections

Key Result

Theorem A

Let $G$ be a cubulated hyperbolic group. Unless $G$ is free or a surface group, $G$ has a one-ended, quasiconvex subgroup of infinite index.

Figures (1)

  • Figure 1: An example of an open star and a link. The example is a closed disc, triangulated as the union of four 2-simplices; the subcomplex $B$ is the vertex in the centre. On the left, the open star is illustrated as the union of the grey open 2-simplices and the black 0- and 1-simplices. On the right, the link is illustrated: it is the union of the 0- and 1-simplices shaded black.

Theorems & Definitions (119)

  • Theorem A
  • Corollary B
  • proof
  • Remark 2
  • Corollary C
  • proof
  • Theorem D
  • Remark 3
  • Theorem E
  • Lemma 1.1
  • ...and 109 more