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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Criticality.CompactDiscreteQuotient
import OAI.Probability.CriticalPercolation.GraphTopology.TopologicalClassificationBoundary
import OAI.Probability.CriticalPercolation.GraphTopology.CompactOrbitKernel
namespace OAI
section
noncomputable section
open Set Function Topology
open scoped Classical Pointwise
namespace CriticalPercolation
namespace CompactQuotient
def FiniteByVirtuallyNilpotent (A : Type*) [Group A] : Prop :=
∃ F : Subgroup A, ∃ hn : F.Normal, let _ := hn
(F : Set A).Finite ∧
∃ N : Subgroup (A ⧸ F), Group.IsNilpotent N ∧ N.index≠0
def UniformFiniteApproximatePowers (A : Type*) [Group A] (K : ℝ) : Prop :=
∃ T : Set A, T.Finite ∧ (1 : A)∈T ∧ T⁻¹=T ∧ Subgroup.closure T=⊤ ∧
∀ R : ℕ, ∃ n≥R, IsApproximateSubgroup K (T^n)
lemma fg_of_uniformFiniteApproximatePowers {A : Type*} [Group A] {K : ℝ}
(h : UniformFiniteApproximatePowers A K) : Group.FG A := by
obtain ⟨T,hT,_,_,hgen,_⟩ := h
exact Group.fg_iff.mpr ⟨T,hgen,hT⟩
end CompactQuotient
namespace GraphTopology
variable {X : Type*} (Γ : SimpleGraph X)
def HasCompactNormalOpenLocallyConnectedSubquotient : Prop :=
let _ : TopologicalSpace X := ⊥
let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ
∃ H : Subgroup (Γ ≃g Γ), ∃ hn : H.Normal, let _ := hn
∃ L : Subgroup (Γ ≃g Γ), L.Normal ∧ H≤L ∧
IsCompact (H : Set (Γ ≃g Γ)) ∧ IsOpen (L : Set (Γ ≃g Γ)) ∧ L.index≠0 ∧
LocallyConnectedSpace (L ⧸ H.subgroupOf L)
def HasFiniteApproximateQuotientCounterexample (K : ℝ) : Prop :=
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)) ∧
Group.FG ((Γ ≃g Γ) ⧸ H) ∧
CompactQuotient.UniformFiniteApproximatePowers ((Γ ≃g Γ) ⧸ H) K ∧
¬CompactQuotient.FiniteByVirtuallyNilpotent ((Γ ≃g Γ) ⧸ H)
theorem classification_failure_dichotomy {K : ℝ}
(hpow : HasUniformCompactApproximatePowers Γ K)
(hfail : ¬HasCompactOpenVirtuallyNilpotentQuotient Γ) :
¬HasCompactNormalOpenLocallyConnectedSubquotient Γ ∨
HasFiniteApproximateQuotientCounterexample Γ K := by
by_cases hcar : HasCompactNormalOpenLocallyConnectedSubquotient Γ
· right
let _ : TopologicalSpace X := ⊥
let _ : DiscreteTopology X := ⟨rfl⟩
let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ
obtain ⟨H,hn,L,_,_,hH,hL,_,hLC⟩ := hcar
let _ := hn
let _ := hLC
let J := orbitKernel Γ H
have hJ := compact_open_orbitKernel_of_open_locallyConnected_subquotient Γ H L hH hL
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 hJ.2 K S hS h1 hinv hgen
intro R
obtain ⟨n,hn,_,_,ha⟩ := hscales R
exact ⟨n,hn,ha⟩
refine ⟨J,inferInstance,hJ.1,hJ.2,
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 hJ.1 hJ.2 F hF N hN)
· exact Or.inl hcar
end GraphTopology
namespace BondGraph
variable {V E : Type*} (G : BondGraph V E)
theorem main_failure_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)) ∧
(¬GraphTopology.HasCompactNormalOpenLocallyConnectedSubquotient
(G.orbitPower o (2*C+1)) ∨
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.classification_failure_dichotomy _ hpow hfail⟩
end BondGraph
end CriticalPercolation
end
end
end OAI
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