| import OAI.RingTheory.DirectFiniteness.ScalarFormulas |
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| namespace OAI |
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| noncomputable section |
| open scoped Classical |
| namespace KaplanskyCounterexample |
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| lemma coprod_fg {G H : Type} [Group G] [Group H] [Group.FG G] [Group.FG H] : |
| Group.FG (Monoid.Coprod G H) := by |
| obtain ⟨α, hα, f, hf⟩ := (Group.fg_iff_exists_freeGroup_hom_surjective_finite (G := G)).mp inferInstance |
| obtain ⟨β, hβ, g, hg⟩ := (Group.fg_iff_exists_freeGroup_hom_surjective_finite (G := H)).mp inferInstance |
| let j : FreeGroup (α ⊕ β) →* Monoid.Coprod G H := |
| FreeGroup.lift (Sum.elim (fun a => Monoid.Coprod.inl (f (FreeGroup.of a))) |
| (fun b => Monoid.Coprod.inr (g (FreeGroup.of b)))) |
| have hl (a : FreeGroup α) : j (FreeGroup.map Sum.inl a) = Monoid.Coprod.inl (f a) := by |
| have he : j.comp (FreeGroup.map Sum.inl) = Monoid.Coprod.inl.comp f := by |
| apply FreeGroup.ext_hom |
| intro x |
| simp [j] |
| exact DFunLike.congr_fun he a |
| have hr (b : FreeGroup β) : j (FreeGroup.map Sum.inr b) = Monoid.Coprod.inr (g b) := by |
| have he : j.comp (FreeGroup.map Sum.inr) = Monoid.Coprod.inr.comp g := by |
| apply FreeGroup.ext_hom |
| intro x |
| simp [j] |
| exact DFunLike.congr_fun he b |
| apply Group.fg_of_surjective (f := j) |
| intro x |
| induction x using Monoid.Coprod.induction_on with |
| | inl a => |
| obtain ⟨a', rfl⟩ := hf a |
| exact ⟨FreeGroup.map Sum.inl a', hl a'⟩ |
| | inr b => |
| obtain ⟨b', rfl⟩ := hg b |
| exact ⟨FreeGroup.map Sum.inr b', hr b'⟩ |
| | mul a b ha hb => |
| obtain ⟨a', rfl⟩ := ha |
| obtain ⟨b', rfl⟩ := hb |
| exact ⟨a' * b', map_mul j a' b'⟩ |
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| lemma multipleHNN_fg {H I : Type} [Group H] [Group.FG H] [Finite I] |
| (A B : I → Subgroup H) (φ : ∀ i, A i ≃* B i) : |
| Group.FG (MultipleHNN.Extension A B φ) := by |
| have : Group.FG (Monoid.Coprod H (FreeGroup I)) := coprod_fg |
| exact Group.fg_of_surjective (f := (MultipleHNN.con A B φ).mk') Con.mk'_surjective |
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| end KaplanskyCounterexample |
| end |
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| end OAI |
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