research/AMR-010-0102.md

metadata

id: AMR-010-0102
classification: SOLVED-IN-LITERATURE
wording_corrected: 'no'

AMR-010-0102 — Finite K(G,1), no Z×Z, balanced ⇒ hyperbolic? (Bestvina Q 1.2)

Problem (corrected statement if needed)

Source: M. Bestvina, Questions in Geometric Group Theory (major revision Aug 2000, updated July 2004), Question 1.2, author-hosted PDF: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf

Q 1.2. Suppose G admits a finite K(G,1), does not contain Z × Z, and whenever x ∈ G is an infinite order element such that x^m and x^n are conjugate, then |m| = |n|. Is G hyperbolic?

The dataset transcription matches the original PDF verbatim (checked against the fetched source); no correction needed. The conjugacy condition is Wise's notion of a balanced group [Wis00, as cited in GKL below]; the two hypotheses together are what Gardam–Kielak–Logan call weakly algebraically hyperbolic (weakly AH). Note that these hypotheses are implied by "G contains no Baumslag–Solitar subgroup BS(m,n)" (Bestvina's Q 1.1): BS(m,n) with |m| ≠ |n| yields conjugate powers of distinct absolute exponents, while BS(m,±m) contains Z² (for |m|=|n| one has [x, y^m] = 1; BS(1,−1) is the Klein bottle group, virtually Z²). So Q 1.2 is a weakening of Q 1.1.

Status / Literature

Resolved — the answer is NO. All items below were verified against the arXiv API or Crossref.

• Italiano, Martelli, Migliorini, Hyperbolic 5-manifolds that fiber over S¹, Invent. Math. 231 (2023), 1–38. DOI: 10.1007/s00222-022-01141-w (verified via Crossref); arXiv: 2105.14795 (verified via arXiv API). They construct finite-volume cusped hyperbolic 5-manifolds fibering over the circle (including the Ratcliffe–Tschantz manifold) and, as a consequence, "build a finite type subgroup of a hyperbolic group that is not hyperbolic" — i.e. a group G of type F (admitting a finite K(G,1)) embedded in a hyperbolic group, hence with no Z×Z and no Baumslag–Solitar subgroups and balanced, which is not hyperbolic. This is exactly a counterexample to Q 1.2 (and to Q 1.1).
• Gardam, Kielak, Logan, Algebraically hyperbolic groups, arXiv: 2112.01331 (v3, 2025; "to appear in Groups, Geometry, and Dynamics" per the arXiv record; verified via arXiv API). Their introduction states explicitly: "Recently, Italiano, Martelli and Migliorini constructed a non-hyperbolic group G of type F that embeds into a hyperbolic group [IMM23, Corollary 2], which is therefore a counter-example to both of Gromov's questions. The group G they construct has geometric and cohomological dimension 4." They attribute the questions, as posed by Gromov, to Bestvina's list [Bes04, Questions 1.1 & 1.2]. They also record that the questions remain open for groups with a finite classifying space of dimension ≤ 3 (their Questions 1.1 and 1.2).
• Brady, Branched coverings of cubical complexes and subgroups of hyperbolic groups, J. London Math. Soc. (2) 60 (1999), 461–480. DOI: 10.1112/s0024610799007644 (verified via Crossref). Gives a finitely presented non-hyperbolic subgroup of a hyperbolic group, showing the "finite K(G,1)" hypothesis cannot be relaxed to "finitely presented" (already noted in Bestvina's remarks to Q 1.1).
• Related companion paper: Italiano–Martelli–Migliorini, Hyperbolic manifolds that fibre algebraically up to dimension 8, J. Inst. Math. Jussieu 23 (2024), 609–646, DOI: 10.1017/s1474748022000536 (verified via Crossref); arXiv:2010.10200.

Positive special cases reported in the literature (as cited in the introduction of Gardam–Kielak–Logan; not independently re-verified by me): yes for 3-manifold groups (via Perelman's geometrization), for free-by-cyclic groups [Brinkmann 2000], for ascending HNN extensions of free groups [Mutanguha 2021], and for fundamental groups of special cube complexes [Caprace–Haglund 2009].

Work done

• Read worklist/AMR-010-0102.md; fetched Bestvina's source PDF and confirmed the transcription of Q 1.2 is verbatim (wording_corrected: no). Note the dataset header "Source item: Question 1.2 (PDF page 2)" matches; Q 1.2 appears on PDF page 2 of the updated list.
• Searched for the current status. Found that the question (and its strengthening Q 1.1) was resolved negatively by Italiano–Martelli–Migliorini; confirmed this via the abstract of arXiv:2105.14795 and, independently, via the explicit statement in Gardam–Kielak–Logan arXiv:2112.01331 (full text read), which also pins down the reference as [IMM23, Corollary 2] and the dimension of the counterexample as 4.
• Verified all primary citations: IMM paper via Crossref (Invent. Math. 231 (2023), 1–38) and arXiv API; Brady 1999 via Crossref; Gardam–Kielak–Logan via arXiv API. (A MathOverflow thread, question 82173, on exactly Q 1.1 exists but could not be fetched — HTTP 403; status confirmed without it.)

Mathematical reasoning (why "embeds in a hyperbolic group" suffices for Q 1.2's hypotheses): Let G be torsion-free and embedded in a hyperbolic group Γ.

• No Z×Z: subgroups of hyperbolic groups contain no Z² (centralizers of infinite-order elements in Γ are virtually cyclic; a Z² would quasi-isometrically embed a Euclidean plane in a δ-hyperbolic space).
• Balancedness: in a hyperbolic group every infinite-order element x has positive translation length τ(x) > 0, translation length is a conjugacy invariant, and τ(x^k) = |k|·τ(x). Hence if x^m and x^n are conjugate, |m|·τ(x) = |n|·τ(x), so |m| = |n|. The same holds in the subgroup G.

Thus the IMM counterexample group — type F, non-hyperbolic, embedded in a (torsion-free) hyperbolic group — satisfies all hypotheses of Q 1.2 while failing the conclusion.

Result

The answer to Q 1.2 is NO. Italiano–Martelli–Migliorini (Invent. Math. 231 (2023), 1–38; arXiv:2105.14795) construct a group G that:

1. admits a finite K(G,1) (is of type F), in fact of geometric and cohomological dimension 4;
2. contains no Z × Z (being a subgroup of a hyperbolic group);
3. is balanced: x^m conjugate to x^n with x of infinite order forces |m| = |n| (translation-length argument, above);
4. is not hyperbolic.

So the class of groups satisfying Bestvina's hypotheses strictly contains the torsion-free hyperbolic groups. The same counterexample simultaneously answers Bestvina's Q 1.1 (no Baumslag–Solitar subgroups) in the negative. The construction uses circle-valued Morse theory/Bestvina–Brady-type finiteness arguments on fibering cusped hyperbolic 5-manifolds; the fiber-kernel (after suitable filling) is the non-hyperbolic type-F subgroup.

What remains

• Low-dimensional case is open: for groups with a finite K(G,1) of dimension ≤ 3 the question is still unresolved — this is precisely Questions 1.1 and 1.2 of Gardam–Kielak–Logan (arXiv:2112.01331). The IMM counterexample has dimension 4, so the dimension bound matters.
• Cohomological dimension 2 (which includes Gersten's question whether every BS-free one-relator group is hyperbolic): open. For cd-2 groups, Gardam–Kielak–Logan prove weakly AH ⇔ BS-free ⇔ "algebraically hyperbolic", so the two versions coincide there.
• Non-positively curved 2-complex case (Bestvina's remark after Q 1.1): if the universal cover contains a flat, must G contain Z×Z? This "flat closing" type question remains open, as does the analogous question for CAT(0) groups in general.
• Structural theory of the enlarged classes: Gardam–Kielak–Logan initiate the study of (weakly) algebraically hyperbolic groups (CSA property, abelian JSJ decompositions); whether a weakly AH group of type F exists that does not embed in any hyperbolic group is open and tied to their Question 1.4 (must a finitely generated cyclic extension of an infinite torsion group have infinite cohomological dimension?).
• Positive answers are known for 3-manifold groups, free-by-cyclic groups, ascending HNN extensions of free groups, and special cube complex groups (cited via Gardam–Kielak–Logan; not independently re-verified).