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feat: add Higman infinite finitely-presented simple group eval problem#384

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feat: add Higman infinite finitely-presented simple group eval problem#384
kim-em merged 2 commits into
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knill-§122-higman-infinite-simple

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@kim-em kim-em commented Jun 2, 2026

This PR adds Higman's infinite finitely-presented simple group theorem as an eval problem: there exists an infinite, finitely presented, simple group — some n and a finite relator set rels ⊆ FreeGroup (Fin n) for which PresentedGroup rels is both IsSimpleGroup and Infinite.

Pure existence of a pathological finite presentation. The first such examples come from Graham Higman's 1974 monograph Finitely Presented Infinite Simple Groups — the Higman–Thompson groups G_{n,r} (for integers n ≥ 2, r ≥ 1) as groups of piecewise-linear bijections of certain Cantor-like spaces. Higman's 1951 paper had earlier constructed a finitely generated infinite simple group as a simple quotient of the four-generator group ⟨a, b, c, d | bab⁻¹ = a², cbc⁻¹ = b², dcd⁻¹ = c², ada⁻¹ = d²⟩ (no nontrivial finite quotients) — but did not give a finite presentation for the simple quotient.

Mathlib v4.30 has FreeGroup, PresentedGroup, IsSimpleGroup, Infinite, the normal-closure substrate (Subgroup.IsNormalClosureFG), HNN extensions, and Britton's lemma (Mathlib/GroupTheory/HNNExtension.lean), but no Higman–Thompson construction, no Higman embedding theorem, and no construction of an infinite finitely-presented simple group on hand.

§122 of Knill's Some Fundamental Theorems in Mathematics (https://people.math.harvard.edu/~knill/graphgeometry/papers/fundamental.pdf).

🤖 Prepared with Claude+Codex+Aristotle

kim-em and others added 2 commits June 2, 2026 05:40
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Codex review of PR #384 confirmed faithfulness of the formal statement
(existential shape correct; IsSimpleGroup includes Nontrivial so
trivial-presentation witnesses are excluded; no vacuous-truth exploit)
and caught two real errors. (1) My prior text claimed Higman's 1951
group was finitely presented and simple; in fact the 1951 paper
constructs a finitely *generated* infinite simple group as a simple
quotient of the four-generator "chain of squarings" group with no
nontrivial finite quotients — the simple quotient need not be finitely
presented. The first infinite finitely-presented simple groups are
the Higman–Thompson groups G_{n,r} from Higman's 1974 monograph.
Rewrote docstring + manifest accordingly. (2) Same HNN/Britton fix as
#383: mathlib v4.30 DOES have HNN extensions and Britton's lemma in
`Mathlib/GroupTheory/HNNExtension.lean'; corrected the "mathlib
lacks X" list.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
@kim-em kim-em merged commit 88267d2 into main Jun 3, 2026
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