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e3e0a0d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 | import OAI.RingTheory.DirectFiniteness.ScalarFormulas
namespace OAI
noncomputable section
open scoped Classical
namespace KaplanskyCounterexample
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'⟩
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
end KaplanskyCounterexample
end
end OAI
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