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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 | import Mathlib
import OAI.Probability.CriticalPercolation.Classification.ClassificationProducerBoundary
import OAI.Probability.CriticalPercolation.GraphGeometry.SubgroupOrbitKernel
import OAI.Probability.CriticalPercolation.Growth.CosetBallIndex
namespace OAI
section
noncomputable section
open Set Function Topology
open scoped Pointwise
namespace CriticalPercolation
namespace CompactQuotient
def HasUniformCompactLocallyConnectedModels (A : Type*) [Group A]
[TopologicalSpace A] (K : ℝ) : Prop :=
∃ B : ℕ, ∀ U : Set A, IsOpen U → IsCompact U → IsApproximateSubgroup K U →
∃ L : Subgroup A, ∃ H : Subgroup L, ∃ hn : H.Normal, let _ := hn
IsCompact (H : Set L) ∧
((Subtype.val : L → A) '' (H : Set L)⊆U^4) ∧
LocallyCompactSpace (L ⧸ H) ∧ LocallyConnectedSpace (L ⧸ H) ∧
∃ F : Finset A, F.card≤B ∧ U⊆(F : Set A)*(L : Set A)
end CompactQuotient
namespace GraphTopology
variable {X : Type*} (Γ : SimpleGraph X)
theorem compact_open_normal_of_raw_models {K : ℝ}
(hpow : HasUniformCompactApproximatePowers Γ K)
(hcar : let _ : TopologicalSpace X := ⊥
let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ
CompactQuotient.HasUniformCompactLocallyConnectedModels (Γ ≃g Γ) K) :
let _ : TopologicalSpace X := ⊥
let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ
∃ J : Subgroup (Γ ≃g Γ), J.Normal ∧
IsCompact (J : Set (Γ ≃g Γ)) ∧ IsOpen (J : Set (Γ ≃g Γ)) := by
let _ : TopologicalSpace X := ⊥
let _ : DiscreteTopology X := ⟨rfl⟩
let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ
obtain ⟨S,hSc,hSo,h1,hinv,hgen,hscales⟩ := hpow
obtain ⟨B,hB⟩ := hcar
obtain ⟨n,hn,hcomp,hopen,happrox⟩ := hscales B
obtain ⟨L,H,hHN,hH,_,hLC,hConn,F,hF,hcover⟩ := hB (S^n) hopen hcomp happrox
let _ := hHN
let _ := hLC
let _ := hConn
let _ : L.FiniteIndex := CosetGrowth.finiteIndex_of_few_cosets S L h1 hinv hgen
n F (hF.trans hn) hcover
exact compact_open_normal_of_finiteIndex_subquotient Γ L H hH
theorem raw_classification_failure_dichotomy {K : ℝ}
(hpow : HasUniformCompactApproximatePowers Γ K)
(hfail : ¬HasCompactOpenVirtuallyNilpotentQuotient Γ) :
(¬(let _ : TopologicalSpace X := ⊥
let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ
CompactQuotient.HasUniformCompactLocallyConnectedModels (Γ ≃g Γ) K)) ∨
HasFiniteApproximateQuotientCounterexample Γ K := by
let _ : TopologicalSpace X := ⊥
let _ : DiscreteTopology X := ⟨rfl⟩
let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ
by_cases hc : CompactQuotient.HasUniformCompactLocallyConnectedModels (Γ ≃g Γ) K
· right
obtain ⟨J,hn,hJc,hJo⟩ := compact_open_normal_of_raw_models Γ hpow hc
let _ := hn
obtain ⟨S,hS,_,h1,hinv,hgen,hscales⟩ := hpow
have hfin : CompactQuotient.UniformFiniteApproximatePowers ((Γ ≃g Γ) ⧸ J) K := by
apply CompactQuotient.finite_generating_approximate_powers J hJo K S hS h1 hinv hgen
intro R
obtain ⟨n,hn,_,_,ha⟩ := hscales R
exact ⟨n,hn,ha⟩
refine ⟨J,hn,hJc,hJo,CompactQuotient.fg_of_uniformFiniteApproximatePowers hfin,hfin,?_⟩
rintro ⟨F,hFN,hF,N,hNil,hN⟩
let _ := hFN
let _ := hNil
exact hfail (CompactQuotient.lift_finite_by_virtually_nilpotent J hJc hJo F hF N hN)
· exact Or.inl hc
end GraphTopology
namespace BondGraph
variable {V E : Type*} (G : BondGraph V E)
theorem main_failure_raw_Carolino_or_finite_BGT [Infinite V]
(hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive)
(hpc : G.criticalProbability<1)
(hper : G.law G.criticalProbability G.percolates≠0) (o : V) :
∃ d : ℕ, 2≤d ∧ ∃ C : ℕ,
(G.orbitPower o (2*C+1)).Connected ∧
(MulAction.orbit G.Aut o).Infinite ∧
(∀ u, ((G.orbitPower o (2*C+1)).neighborSet u).Finite) ∧
(∀ u v, ∃ g : G.orbitPower o (2*C+1) ≃g G.orbitPower o (2*C+1), g u=v) ∧
(∀ R : ℕ, ∃ n≥R, ∀ u : MulAction.orbit G.Aut o,
(GraphTopology.reachBall (G.orbitPower o (2*C+1)) u n).ncard≤n^(d+1)) ∧
GraphTopology.HasUniformCompactApproximatePowers
(G.orbitPower o (2*C+1)) (5^(d+2)) ∧
((¬(let _ : TopologicalSpace (MulAction.orbit G.Aut o) := ⊥
let _ : TopologicalSpace (G.orbitPower o (2*C+1) ≃g G.orbitPower o (2*C+1)) :=
GraphTopology.autTopology (G.orbitPower o (2*C+1))
CompactQuotient.HasUniformCompactLocallyConnectedModels
(G.orbitPower o (2*C+1) ≃g G.orbitPower o (2*C+1)) (5^(d+2)))) ∨
GraphTopology.HasFiniteApproximateQuotientCounterexample
(G.orbitPower o (2*C+1)) (5^(d+2))) := by
obtain ⟨d,hd,C,hconn,hinf,hloc,ht,hpoly,hpow,hfail⟩ :=
G.main_failure_topological_classification_counterexample hc hl hq hpc hper o
exact ⟨d,hd,C,hconn,hinf,hloc,ht,hpoly,hpow,
GraphTopology.raw_classification_failure_dichotomy _ hpow hfail⟩
end BondGraph
end CriticalPercolation
end
end
end OAI
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