research/AMR-010-0107.md

---
id: AMR-010-0107
classification: OPEN-TRIAGE
wording_corrected: no
---

# AMR-010-0107 — Gromov: hyperbolic groups of dimension n with all infinite-index subgroups free

## Problem (corrected statement if needed)

The dataset transcription matches the source verbatim; no correction was needed.
Original wording from M. Bestvina, *Questions in Geometric Group Theory*
(major revision Aug 2000, updated July 2004), Question 1.7, Section 1.2
"Subgroups of Hyperbolic Groups" (author-hosted PDF,
<https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf>, fetched
and read directly):

> **Q 1.7. (Gromov)** For a given n is there an example of a hyperbolic group of
> dimension n in which every infinite index subgroup is free? Or in which there
> are no (quasi-convex) subgroups with codimension ≤ k for a given k ≤ n−2.

Here "dimension" is understood as (virtual/rational) cohomological dimension;
"codimension" is not defined in the list, and is plausibly meant either as
cd(G) − cd(H) or in Sageev's sense (relative ends / limit-set codimension).
The question is the "opposite possibility" to Gromov's Q 1.6 (does every
1-ended hyperbolic group contain a surface subgroup?). The closely related
Q 1.11 (Whyte) — *can a 1-ended hyperbolic group that is not virtually a
surface group have every infinite-index subgroup free?* — is essentially the
n = 2 sharpened form of the same question.

## Status / Literature

All items below were verified against Crossref or the arXiv API during this
review.

- **n = 1, 2: examples exist (classical).** Free groups (n = 1) by
  Nielsen–Schreier. Closed hyperbolic surface groups (n = 2): every
  infinite-index subgroup of a surface group is free (attributed to Johansson;
  the modern homological proof is Strebel's theorem that infinite-index
  subgroups of PD²-groups have cd ≤ 1, hence are free by Stallings–Swan):
  R. Strebel, *A remark on subgroups of infinite index in Poincaré duality
  groups*, Comment. Math. Helv. 52 (1977), 317–324,
  DOI 10.1007/BF02567371 (verified via Crossref).

- **Strong negative result in the cubulated case (the main recent progress).**
  H. Wilton, *Surface groups among cubulated hyperbolic and one-relator
  groups*, arXiv:2406.02121 (v3, Jan 2026, "final version accepted for
  publication"; verified via arXiv API and by reading the HTML full text).
  Theorem A: *a cubulated hyperbolic group G has a one-ended quasiconvex
  subgroup of infinite index unless G is free or a surface group.* The author
  states explicitly that this "answers questions of Gromov and Whyte in the
  cubulated case [Bestvina's list, Questions 1.7 and 1.11]". Since cubulated
  hyperbolic groups include C′(1/6) small-cancellation groups and, by the
  Agol–Wise virtual Haken theory, all closed hyperbolic 3-manifold groups,
  **no cubulated hyperbolic group of dimension ≥ 3 answers Q 1.7**. Theorem D
  gives the analogous statement for one-relator groups (subgroup produced may
  be infinitely generated). Wilton's Question 0.1 records the fully general
  finitely-presented version as open, and his §5/§6 record that the
  higher-dimensional (cd ≥ 3) Strebel-type picture is unresolved.

- **Two-generator one-relator case.**
  G. Gardam, D. Kielak, A. D. Logan, *The Surface Group Conjectures for
  groups with two generators*, arXiv:2202.11093 (verified via arXiv API):
  a two-generator one-relator group with every infinite-index subgroup free is
  free or a surface group.

- **n = 3, manifold groups ruled out.**
  J. Kahn, V. Marković, *Immersing almost geodesic surfaces in a closed
  hyperbolic three manifold*, Ann. of Math. 175 (2012), 1127–1190,
  DOI 10.4007/annals.2012.175.3.4 (verified via Crossref): every closed
  hyperbolic 3-manifold group contains a (quasiconvex) surface subgroup, so
  closed hyperbolic 3-manifold groups never answer Q 1.7 for n = 3.

- **Background for the codimension clause.**
  M. Kapovich, B. Kleiner, *Coarse Alexander duality and duality groups*,
  J. Differential Geom. 69 (2005), 279–352, DOI 10.4310/jdg/1121449108
  (verified via Crossref): for a quasiconvex subgroup H of a hyperbolic group
  G, the homology of the limit set ΛH and the topology of its complement in
  ∂G are related by coarse Alexander duality; this is the standard tool for
  making "codimension of a quasiconvex subgroup" precise. Codimension-1
  quasiconvex subgroups are tied to cubulations (Sageev's construction); this
  Sageev–Niblo–Roller theory is cited here from general knowledge, not
  independently re-verified in this review.

## Work done

- Fetched and read the source PDF (Bestvina's updated questions list);
  confirmed the dataset transcription is exact, including the trailing clause
  "for a given k ≤ n−2".
- Searched the web for the status of Q 1.7; identified Wilton's 2024–2026
  paper as the decisive recent development and verified it (arXiv API record
  plus reading the introduction of the HTML version, which explicitly cites
  Bestvina's Questions 1.7 and 1.11 as being answered in the cubulated case).
- Verified Gardam–Kielak–Logan (arXiv API), Strebel 1977, Kahn–Marković 2012,
  Kapovich–Kleiner 2005 (all via Crossref/arXiv API; one initially guessed
  DOI for Strebel was wrong — it resolved to a Kervaire–Murthy paper — and was
  corrected via a Crossref bibliographic query).
- Elementary deductions constraining any example G of dimension n ≥ 2 with all
  infinite-index subgroups free (pure reasoning, no literature needed):
  1. **G is torsion-free.** Every finite subgroup has infinite index (G is
     infinite), hence must be free, hence trivial.
  2. **G is freely indecomposable and 1-ended.** If G splits over a finite
     subgroup, the vertex groups have infinite index, hence are free; a
     graph of free groups with finite edge groups is virtually free, so
     cd(G) ≤ 1, contradicting n ≥ 2. (Virtually-cyclic is likewise excluded.)
  3. Consequently cd(G) equals the geometric dimension, G is a torsion-free
     1-ended hyperbolic group, and every infinite-index subgroup has cd ≤ 1.
     The question is thus precisely: does such a group exist in cd ≥ 3, i.e.
     is there a "higher-dimensional Strebel phenomenon" beyond PD²-groups?
  4. For the codimension clause with k = 1: hyperbolic groups with Kazhdan's
     property (T) (e.g. cocompact lattices in Sp(n,1)) admit no proper
     codimension-1 subgroups in Sageev's sense, since a codimension-1
     subgroup yields a nontrivial action on a CAT(0) cube complex and
     property (T) forces a fixed point (Sageev/Niblo–Roller theory;
     cited from background knowledge, not re-verified here). So the k = 1
     case of the second clause is essentially known, under that
     interpretation of "codimension".
- Combining (3) with Wilton's Theorem A and Agol's theorem (cubulated
  hyperbolic ⟹ virtually special): **any example for n ≥ 3 must be a
  hyperbolic group with no proper cocompact cubulation** — a class that
  includes property-(T) hyperbolic groups and various non-cubulated
  quotients, about whose subgroup structure very little is known.

## Result

The problem is **open**, with a sharp literature triage:

- n = 1 (free groups) and n = 2 (closed hyperbolic surface groups) are the
  only known examples of hyperbolic groups of dimension n with every
  infinite-index subgroup free.
- For n ≥ 3 the answer is negative in every class where the question is
  understood: cubulated hyperbolic groups (Wilton, arXiv:2406.02121, which
  covers closed hyperbolic 3-manifold groups and small-cancellation groups),
  one-relator groups (Wilton's Theorem D; Gardam–Kielak–Logan for two
  generators), and closed hyperbolic 3-manifold groups independently
  (Kahn–Marković surface subgroups).
- No construction of an n ≥ 3 example exists anywhere in the literature, and
  Wilton explicitly records the general question (his Question 0.1, and the
  cd ≥ 3 variants in his §6) as open. The codimension clause is likewise open
  in general (only the k = 1 case is settled, via property (T), under the
  Sageev interpretation).

No solution or new theorem is claimed here; the contribution is the verified
triage plus the elementary structural constraints (torsion-free, 1-ended,
non-cubulated) on any hypothetical example.

## What remains

- **Core open case:** does there exist a hyperbolic group G with cd(G) = n ≥ 3
  (equivalently dim ∂G = n − 1 ≥ 2) whose infinite-index subgroups are all
  free? By the constraints above, any example must be torsion-free, 1-ended,
  and admit no proper cocompact action on a CAT(0) cube complex — so the
  question is a stress test for the reach of cubulation techniques, and a
  negative answer in general would likely require extending Wilton's
  Whitehead-complex/cut-width machinery beyond the cubulated world, which
  Wilton himself describes as "well beyond current technology".
- **Codimension clause:** for 2 ≤ k ≤ n − 2, does there exist a hyperbolic
  group of dimension n with no quasiconvex subgroup of codimension ≤ k?
  Nothing in the verified literature settles this; coarse Alexander duality
  (Kapovich–Kleiner) is the natural framework, and the surface-subgroup
  problem for higher-rank/rank-one lattices (e.g. Sp(n,1)) is a key test
  case.
- Natural next steps: (a) decide the question for property-(T) hyperbolic
  groups (do cocompact lattices in Sp(n,1) or their small-cancellation
  quotients contain infinite-index non-free — e.g. surface — subgroups?);
  (b) extend the "strong Strebel" converse of Wilton's §5–6 to cd = 3 for
  arbitrary (non-cubulated) hyperbolic groups; (c) check whether any
  hyperbolic group with Menger-curve or Sierpiński boundary of dim ≥ 2 can
  have all infinite-index subgroups free.