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f205175 | 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 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 | import Mathlib
namespace OAI
section
noncomputable section
open Set Function Topology
open scoped Classical Pointwise
namespace CriticalPercolation.CompactQuotient
variable {A : Type*} [Group A] [TopologicalSpace A] [IsTopologicalGroup A]
variable (H : Subgroup A) [H.Normal]
lemma compact_preimage_of_finite (hH : IsCompact (H : Set A))
(T : Set (A ⧸ H)) (hT : T.Finite) : IsCompact ((QuotientGroup.mk' H) ⁻¹' T) := by
let a : A ⧸ H → A := fun t => (QuotientGroup.mk_surjective t).choose
have ha : ∀ t, QuotientGroup.mk' H (a t)=t := fun t =>
(QuotientGroup.mk_surjective t).choose_spec
have he : (QuotientGroup.mk' H) ⁻¹' T =
⋃ t∈T, (fun h : A => a t*h) '' (H : Set A) := by
ext g
constructor
· intro hg
refine Set.mem_iUnion.mpr ⟨QuotientGroup.mk' H g,Set.mem_iUnion.mpr ⟨hg,?_⟩⟩
refine ⟨(a (QuotientGroup.mk' H g))⁻¹*g,?_,by simp⟩
apply (QuotientGroup.eq_one_iff _).mp
change QuotientGroup.mk' H ((a (QuotientGroup.mk' H g))⁻¹*g)=1
rw [map_mul,map_inv,ha,inv_mul_cancel]
· intro hg
obtain ⟨t,hut⟩ := Set.mem_iUnion.mp hg
obtain ⟨ht,h,hHh,rfl⟩ := Set.mem_iUnion.mp hut
change QuotientGroup.mk' H (a t*h)∈T
have hh : QuotientGroup.mk' H h=1 := (QuotientGroup.eq_one_iff _).mpr hHh
rw [map_mul,ha,hh,mul_one]
exact ht
rw [he]
exact hT.isCompact_biUnion fun t _ => hH.image (continuous_const.mul continuous_id)
theorem compact_open_comap_finite
(hcomp : IsCompact (H : Set A)) (hopen : IsOpen (H : Set A))
(F : Subgroup (A ⧸ H)) (hF : (F : Set (A ⧸ H)).Finite) :
IsCompact (F.comap (QuotientGroup.mk' H) : Set A) ∧
IsOpen (F.comap (QuotientGroup.mk' H) : Set A) := by
let _ : DiscreteTopology (A ⧸ H) := QuotientGroup.discreteTopology hopen
exact ⟨compact_preimage_of_finite H hcomp F hF,
(isOpen_discrete (F : Set (A ⧸ H))).preimage QuotientGroup.continuous_mk⟩
theorem finite_generating_approximate_powers
(hopen : IsOpen (H : Set A)) (K : ℝ) (S : Set A)
(hcomp : IsCompact S) (hone : (1 : A)∈S) (hsym : S⁻¹=S)
(hgen : Subgroup.closure S=⊤)
(hpow : ∀ R : ℕ, ∃ n≥R, IsApproximateSubgroup K (S^n)) :
∃ T : Set (A ⧸ H), T.Finite ∧ (1 : A ⧸ H)∈T ∧ T⁻¹=T ∧
Subgroup.closure T=⊤ ∧
∀ R : ℕ, ∃ n≥R, IsApproximateSubgroup K (T^n) := by
let _ : DiscreteTopology (A ⧸ H) := QuotientGroup.discreteTopology hopen
refine ⟨(QuotientGroup.mk' H) '' S,
isCompact_iff_finite.mp (hcomp.image QuotientGroup.continuous_mk),
⟨1,hone,map_one _⟩,?_,?_,?_⟩
· rw [←Set.image_inv,hsym]
· rw [←MonoidHom.map_closure,hgen]
exact Subgroup.map_top_of_surjective _ QuotientGroup.mk_surjective
· intro R
obtain ⟨n,hn,ha⟩ := hpow R
refine ⟨n,hn,?_⟩
rw [←Set.image_pow]
exact ha.image (QuotientGroup.mk' H)
theorem lift_finite_by_virtually_nilpotent
(hcomp : IsCompact (H : Set A)) (hopen : IsOpen (H : Set A))
(F : Subgroup (A ⧸ H)) [F.Normal] (hF : (F : Set (A ⧸ H)).Finite)
(N : Subgroup ((A ⧸ H) ⧸ F)) [Group.IsNilpotent N] (hN : N.index≠0) :
∃ C : Subgroup A, ∃ hn : C.Normal, let _ := hn
IsCompact (C : Set A) ∧ IsOpen (C : Set A) ∧
∃ M : Subgroup (A ⧸ C), Group.IsNilpotent M ∧ M.index≠0 := by
let f : A →* (A ⧸ H) ⧸ F := (QuotientGroup.mk' F).comp (QuotientGroup.mk' H)
have hf : Surjective f := QuotientGroup.mk_surjective.comp QuotientGroup.mk_surjective
have hker : f.ker=F.comap (QuotientGroup.mk' H) := by
change ((QuotientGroup.mk' F).comp (QuotientGroup.mk' H)).ker=_
rw [←MonoidHom.comap_ker,QuotientGroup.ker_mk']
let e : A ⧸ f.ker ≃* (A ⧸ H) ⧸ F := QuotientGroup.quotientKerEquivOfSurjective f hf
let M : Subgroup (A ⧸ f.ker) := N.comap e.toMonoidHom
have hnil : Group.IsNilpotent M := by
let j : M →* N :=
{toFun := fun g => ⟨e g.val,g.property⟩
map_one' := Subtype.ext (map_one e)
map_mul' := fun g h => Subtype.ext (map_mul e g.val h.val)}
have hj : Injective j := by
intro g h he
apply Subtype.ext
exact e.injective (congrArg Subtype.val he)
apply Subgroup.isNilpotent_of_ker_le_center j
rw [(MonoidHom.ker_eq_bot_iff j).mpr hj]
exact bot_le
have hi : M.index=N.index := N.index_comap_of_surjective e.surjective
have hc := compact_open_comap_finite H hcomp hopen F hF
rw [←hker] at hc
exact ⟨f.ker,inferInstance,hc.1,hc.2,M,hnil,hi ▸ hN⟩
end CriticalPercolation.CompactQuotient
end
end
end OAI
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