File size: 3,060 Bytes
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 | import Mathlib
import OAI.Probability.CriticalPercolation.Classification.ClassificationProducerBoundary
import OAI.Probability.CriticalPercolation.Classification.FiniteBGTBridge
namespace OAI
section
noncomputable section
open Set Function Topology
open scoped Pointwise
namespace CriticalPercolation.CompactQuotient
variable {A : Type*} [Group A]
def FiniteBGTPowerOutput (S : Set A) (n : ℕ) : Prop :=
∃ H N : Subgroup A, ∃ hH : H.Normal, let _ := hH
N.Normal ∧ H≤N ∧ (H : Set A)⊆S^(4*n) ∧ N.index≠0 ∧
Group.IsNilpotent (N ⧸ H.subgroupOf N)
def HasUniformFiniteBGTModels (A : Type*) [Group A] (K : ℝ) : Prop :=
∃ R : ℕ, ∀ S : Set A, S.Finite → (1 : A)∈S → S⁻¹=S →
Subgroup.closure S=⊤ → ∀ n≥R, IsApproximateSubgroup K (S^n) →
FiniteBGTPowerOutput S n
lemma finiteByVirtuallyNilpotent_of_finiteBGTPowerOutput
(S : Set A) (hS : S.Finite) (n : ℕ) (h : FiniteBGTPowerOutput S n) :
FiniteByVirtuallyNilpotent A := by
obtain ⟨H,N,hn,_,hHN,hH,hN,hNil⟩ := h
let _ := hn
let _ := hNil
exact ⟨H,hn,finite_by_nilpotent_of_BGT_output S hS n H N hHN hH hN⟩
lemma one_le_of_uniformFiniteApproximatePowers {K : ℝ}
(h : UniformFiniteApproximatePowers A K) : 1≤K := by
obtain ⟨S,_,_,_,_,hpow⟩ := h
obtain ⟨n,_,hn⟩ := hpow 0
exact hn.one_le
theorem finiteByVirtuallyNilpotent_of_uniform_models {K : ℝ}
(hpow : UniformFiniteApproximatePowers A K)
(hbgt : HasUniformFiniteBGTModels A K) : FiniteByVirtuallyNilpotent A := by
obtain ⟨R,hR⟩ := hbgt
obtain ⟨S,hS,h1,hinv,hgen,hscales⟩ := hpow
obtain ⟨n,hn,ha⟩ := hscales R
exact finiteByVirtuallyNilpotent_of_finiteBGTPowerOutput S hS n
(hR S hS h1 hinv hgen n hn ha)
theorem no_uniform_finite_BGT_models {K : ℝ}
(hpow : UniformFiniteApproximatePowers A K)
(hf : ¬FiniteByVirtuallyNilpotent A) : ¬HasUniformFiniteBGTModels A K :=
fun h => hf (finiteByVirtuallyNilpotent_of_uniform_models hpow h)
end CriticalPercolation.CompactQuotient
namespace CriticalPercolation.GraphTopology
variable {X : Type*} (Γ : SimpleGraph X)
theorem finite_classification_obstruction {K : ℝ}
(h : HasFiniteApproximateQuotientCounterexample Γ K) :
let _ : TopologicalSpace X := ⊥
let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ
∃ H : Subgroup (Γ ≃g Γ), ∃ hn : H.Normal, let _ := hn
IsCompact (H : Set (Γ ≃g Γ)) ∧ IsOpen (H : Set (Γ ≃g Γ)) ∧
1≤K ∧ Group.FG ((Γ ≃g Γ) ⧸ H) ∧
CompactQuotient.UniformFiniteApproximatePowers ((Γ ≃g Γ) ⧸ H) K ∧
¬CompactQuotient.HasUniformFiniteBGTModels ((Γ ≃g Γ) ⧸ H) K := by
let _ : TopologicalSpace X := ⊥
let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ
obtain ⟨H,hn,hc,ho,hfg,hpow,hfail⟩ := h
let _ := hn
exact ⟨H,hn,hc,ho,CompactQuotient.one_le_of_uniformFiniteApproximatePowers hpow,
hfg,hpow,CompactQuotient.no_uniform_finite_BGT_models hpow hfail⟩
end CriticalPercolation.GraphTopology
end
end
end OAI
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