HyperbolicGroupsResiduallyFinite.lean`

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3 declarations, 3 contain sorry, 0 declared as axioms

Residual finiteness of hyperbolic groups

Is every hyperbolic group residually finite? Kapovich and Wise [KaWi00] proved that this question is equivalent to asking whether every nontrivial hyperbolic group has a proper subgroup of finite index. Both questions were answered negatively by an internal OpenAI model in September 2026 [OAI26a].

References:

[BMS] G. Baumslag, A. G. Myasnikov and V. Shpilrain, Open problems in combinatorial group theory, Problem (H1), online.
[Be04] M. Bestvina, Questions in geometric group theory, updated July 2004, Q 1.15, pdf.
[KaWi00] I. Kapovich and D. T. Wise, The equivalence of some residual properties of word-hyperbolic groups, J. Algebra 223 (2000), 562–583, doi:10.1006/jabr.1999.8104.
[OAI26a] OpenAI, A torsion-free hyperbolic group that is not residually finite. OpenAI Math Release preprint (2026). https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/a-torsion-free-hyperbolic-group-that-is-not-residually-finite-September-23-2026/paper.pdf

Imports (1)

FormalConjecturesUtil

theorem HyperbolicGroupsResiduallyFinite.hyperbolic_residuallyFinite Is every hyperbolic group residually finite? sorry line 59 source
Source documentation
Is every hyperbolic group residually finite?
This is Problem (H1)(a) in [BMS] and Q 1.15 in [Be04].
No. This was disproved by an internal OpenAI model in September 2026 [OAI26a], which constructs a torsion-free hyperbolic group that is not residually finite. The Lean proof of [OAI26a] defines hyperbolicity by uniformly thin geodesic triangles in a Cayley graph. That definition is equivalent to Group.IsHyperbolic, but the equivalence is not part of the Lean proof.

L4DEL explanation
Is every hyperbolic group residually finite? This is Problem (H1)(a) in [BMS] and Q 1.15 in [Be04]. No. This was disproved by an internal OpenAI model in September 2026 [OAI26a], which constructs a torsion-free hyperbolic group that is not residually finite. The Lean proof of [OAI26a] defines hyperbolicity by uniformly thin geodesic triangles in a Cayley graph. That definition is equivalent to Group.IsHyperbolic, but the equivalence is not part of the Lean proof.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Iff, False, Group, Group.IsHyperbolic, Group.ResiduallyFinite
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem hyperbolic_residuallyFinite : answer(False) ↔
    ∀ (G : Type) [Group G] [Group.IsHyperbolic G], Group.ResiduallyFinite G := by
…
```
Mentions: Group.IsHyperbolic
theorem HyperbolicGroupsResiduallyFinite.hyperbolic_exists_finiteIndex_ne_top Does every nontrivial hyperbolic group have a proper subgroup of finite index? sorry line 73 source
Source documentation
Does every nontrivial hyperbolic group have a proper subgroup of finite index?
This is Problem (H1)(b) in [BMS]. The trivial group has no proper subgroup, so we assume that $G$ is nontrivial.
No. By [KaWi00] (see hyperbolic_residuallyFinite_iff_exists_finiteIndex_ne_top), this problem is equivalent to hyperbolic_residuallyFinite, which was disproved in [OAI26a].

depends on sorry what this means
Statement uses: Iff, False, Group, Group.IsHyperbolic, Nontrivial, Exists, Subgroup, And, Ne, Top.top, Subgroup.FiniteIndex
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem hyperbolic_exists_finiteIndex_ne_top : answer(False) ↔
    ∀ (G : Type) [Group G] [Group.IsHyperbolic G] [Nontrivial G],
      ∃ H : Subgroup G, H ≠ ⊤ ∧ H.FiniteIndex := by
…
```
Mentions: Group.IsHyperbolic
theorem HyperbolicGroupsResiduallyFinite.hyperbolic_residuallyFinite_iff_exists_finiteIndex_ne_top Every hyperbolic group is residually finite if and only if every nontrivial hyperbolic group has a proper subgroup of finite index sorry line 85 source
Source documentation
Every hyperbolic group is residually finite if and only if every nontrivial hyperbolic group has a proper subgroup of finite index.
This is the theorem of [KaWi00] that Problems (H1)(a) and (H1)(b) are equivalent.

depends on sorry what this means
Statement uses: Iff, Group, Group.IsHyperbolic, Group.ResiduallyFinite, Nontrivial, Exists, Subgroup, And, Ne, Top.top, Subgroup.FiniteIndex
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem hyperbolic_residuallyFinite_iff_exists_finiteIndex_ne_top :
    (∀ (G : Type) [Group G] [Group.IsHyperbolic G], Group.ResiduallyFinite G) ↔
      ∀ (G : Type) [Group G] [Group.IsHyperbolic G] [Nontrivial G],
        ∃ H : Subgroup G, H ≠ ⊤ ∧ H.FiniteIndex := by
…
```
Mentions: Group.IsHyperbolic