| import Mathlib |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.TranslatedDisjoint |
|
|
| namespace OAI |
|
|
| section |
| section |
| open scoped symmDiff |
| namespace SimpleAmenable |
| open scoped commutatorElement |
| open scoped commutatorElement |
| section AlignedGroups |
|
|
| variable {A H : Type*} [Group A] [Group H] {B : Type*} |
|
|
| noncomputable def alignedGroup (t : A →* H) (x : B → H) (P : B → Prop) : Subgroup H := |
| Subgroup.closure {y | ∃ b, P b ∧ ∃ u, y = t u*x b*(t u)⁻¹} |
|
|
| theorem alignedGroup_generator (t : A →* H) (x : B → H) (P : B → Prop) |
| (b : B) (hb : P b) (u : A) : t u*x b*(t u)⁻¹ ∈ alignedGroup t x P := |
| Subgroup.subset_closure ⟨b,hb,u,rfl⟩ |
|
|
| theorem alignedGroup_input (t : A →* H) (x : B → H) (P : B → Prop) |
| (b : B) (hb : P b) : x b ∈ alignedGroup t x P := by |
| simpa only [map_one,inv_one,one_mul,mul_one] using alignedGroup_generator t x P b hb 1 |
|
|
| theorem alignedGroup_conjugate (t : A →* H) (x : B → H) (P : B → Prop) |
| (u : A) {y : H} (hy : y ∈ alignedGroup t x P) : |
| t u*y*(t u)⁻¹ ∈ alignedGroup t x P := by |
| let f : H →* H := (MulAut.conj (t u)).toMonoidHom |
| have hh : alignedGroup t x P ≤ (alignedGroup t x P).comap f := by |
| apply (Subgroup.closure_le _).mpr |
| rintro z ⟨b,hb,v,rfl⟩ |
| change t u*(t v*x b*(t v)⁻¹)*(t u)⁻¹ ∈ alignedGroup t x P |
| have he : t u*(t v*x b*(t v)⁻¹)*(t u)⁻¹ = t (u*v)*x b*(t (u*v))⁻¹ := by |
| rw [map_mul]; group |
| rw [he] |
| exact alignedGroup_generator t x P b hb (u*v) |
| exact hh hy |
|
|
| theorem alignedGroup_commute (t : A →* H) (x : B → H) (P Q : B → Prop) |
| (hc : ∀ b, P b → ∀ c, Q c → ∀ u v, |
| Commute (t u*x b*(t u)⁻¹) (t v*x c*(t v)⁻¹)) : |
| ∀ g ∈ alignedGroup t x P, ∀ h ∈ alignedGroup t x Q, Commute g h := by |
| have hh : alignedGroup t x P ≤ Subgroup.centralizer (alignedGroup t x Q) := by |
| apply (Subgroup.closure_le _).mpr |
| rintro g ⟨b,hb,u,rfl⟩ |
| rw [alignedGroup,Subgroup.centralizer_closure] |
| intro h hh |
| obtain ⟨c,hc',v,rfl⟩ := hh |
| exact (hc b hb c hc' u v).symm.eq |
| intro g hg h hmem |
| exact (hh hg h hmem).symm |
|
|
| theorem alignedGroup_map {G : Type*} [Group G] (q : H →* G) |
| (t : A →* H) (x : B → H) (P : B → Prop) : |
| (alignedGroup t x P).map q = alignedGroup (q.comp t) (fun b => q (x b)) P := by |
| rw [alignedGroup,MonoidHom.map_closure] |
| congr 1 |
| ext y |
| constructor |
| · rintro ⟨z,⟨b,hb,u,rfl⟩,rfl⟩ |
| exact ⟨b,hb,u,by simp⟩ |
| · rintro ⟨b,hb,u,rfl⟩ |
| exact ⟨t u*x b*(t u)⁻¹,⟨b,hb,u,rfl⟩,by simp⟩ |
|
|
| end AlignedGroups |
|
|
| section SourceAlignedGroups |
|
|
| def SmallConditional (m : ℕ) := Fin 5 × {σ : Equiv.Perm (Fin (m+1)) // |
| σ ∈ alternatingGroup (Fin (m+1)) ∧ σ.support.card ≤ 5} |
|
|
| instance (m : ℕ) : Finite (SmallConditional m) := by unfold SmallConditional; infer_instance |
|
|
| noncomputable def sourceAlignedGroup (a : ℕ) (r : CutRing) (m : ℕ) (hm : 2 ≤ m) |
| (M : ℕ) (t : Multiplicative (FreeAbelianGroup (Fin m × Fin 2)) →* |
| BoundedRelationCover M (alternatingGenerator a r m hm)) (I : Finset (Fin (m+1))) : |
| Subgroup (BoundedRelationCover M (alternatingGenerator a r m hm)) := |
| alignedGroup t (fun b : SmallConditional m => PresentedGroup.of (Sum.inl b)) |
| (fun b => b.2.val.support ⊆ I) |
|
|
| theorem source_aligned_disjoint_eventually (a : ℕ) (r : CutRing) (m : ℕ) |
| (hm : 2 ≤ m) (hm' : 10 < m+1) : |
| ∃ L : ℕ, ∀ M : ℕ, L ≤ M → |
| ∃ t : Multiplicative (FreeAbelianGroup (Fin m × Fin 2)) →* |
| BoundedRelationCover M (alternatingGenerator a r m hm), |
| (∀ i, t (Multiplicative.ofAdd (FreeAbelianGroup.of i)) = PresentedGroup.of (Sum.inr i)) ∧ |
| (coverMap M (alternatingGenerator a r m hm)).comp t = sourceLatticeMap a r m hm ∧ |
| ∀ (I J : Finset (Fin (m+1))), Disjoint I J → |
| ∀ g ∈ sourceAlignedGroup a r m hm M t I, |
| ∀ h ∈ sourceAlignedGroup a r m hm M t J, Commute g h := by |
| obtain ⟨L,hL⟩ := source_translated_disjoint_eventually a r m hm hm' |
| refine ⟨L,fun M hM => ?_⟩ |
| obtain ⟨t,ht,htq,hc⟩ := hL M hM |
| refine ⟨t,ht,htq,?_⟩ |
| intro I J hIJ |
| apply alignedGroup_commute |
| intro b hb c hc' u v |
| exact hc b.1 c.1 b.2 c.2 (hIJ.mono hb hc') u v |
|
|
| end SourceAlignedGroups |
|
|
| end SimpleAmenable |
| end |
| end |
|
|
| end OAI |
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|
