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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Growth.GraphQuotientGrowth
import OAI.Probability.CriticalPercolation.Classification.QualitativeClassificationBoundary
namespace OAI
section
noncomputable section
open Set Function Topology
open scoped Classical Pointwise
namespace CriticalPercolation
namespace CompactQuotient
def HasSparsePolynomialWordGrowth (A : Type*) [Group A] : Prop :=
∃ S : Set A, S.Finite ∧ (1 : A)∈S ∧ S⁻¹=S ∧ Subgroup.closure S=⊤ ∧
∃ d : ℕ, ∀ R : ℕ, ∃ n≥R, (S^n).ncard≤n^d
end CompactQuotient
namespace GraphTopology
variable {X : Type*} (Γ : SimpleGraph X)
theorem quotient_sparse_polynomial_word_growth
[TopologicalSpace X] [DiscreteTopology X]
(hc : Γ.Connected) (hl : ∀ x, (Γ.neighborSet x).Finite)
(ht : ∀ u v, ∃ g : Γ ≃g Γ, g u=v) (o : X) (d : ℕ)
(hpoly : ∀ R : ℕ, ∃ n≥R, (reachBall Γ o n).ncard≤n^d)
(H : Subgroup (Γ ≃g Γ)) [H.Normal]
(ho : @IsOpen (Γ ≃g Γ) (autTopology Γ) (H : Set (Γ ≃g Γ))) :
CompactQuotient.HasSparsePolynomialWordGrowth ((Γ ≃g Γ) ⧸ H) := by
obtain ⟨M,_,hT,h1,hinv,hgen,hncard⟩ :=
quotient_powers_ncard_le Γ hc hl ht o H ho
refine ⟨(QuotientGroup.mk' H) '' displacement Γ o 1,hT,h1,hinv,hgen,d+1,?_⟩
intro R
obtain ⟨n,hn,hp⟩ := hpoly (max R (max M 1))
have hnR : R≤n := (le_max_left _ _).trans hn
have hnM : M≤n := (le_max_left _ _).trans ((le_max_right _ _).trans hn)
have hn1 : 1≤n := (le_max_right _ _).trans ((le_max_right _ _).trans hn)
refine ⟨n,hnR,?_⟩
calc
(((QuotientGroup.mk' H) '' displacement Γ o 1)^n).ncard ≤
(reachBall Γ o n).ncard*M := hncard n hn1
_ ≤ n^d*M := Nat.mul_le_mul_right M hp
_ ≤ n^d*n := Nat.mul_le_mul_left _ hnM
_ = n^(d+1) := (pow_succ n d).symm
theorem weak_Gromov_counterexample_of_finite_classification_failure {K : ℝ}
(hc : Γ.Connected) (hl : ∀ x, (Γ.neighborSet x).Finite)
(ht : ∀ u v, ∃ g : Γ ≃g Γ, g u=v) (o : X) (d : ℕ)
(hpoly : ∀ R : ℕ, ∃ n≥R, (reachBall Γ o n).ncard≤n^d)
(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)) ∧
Group.FG ((Γ ≃g Γ) ⧸ H) ∧
CompactQuotient.HasSparsePolynomialWordGrowth ((Γ ≃g Γ) ⧸ H) ∧
¬Group.IsVirtuallyNilpotent ((Γ ≃g Γ) ⧸ H) := by
let _ : TopologicalSpace X := ⊥
let _ : DiscreteTopology X := ⟨rfl⟩
let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ
obtain ⟨H,hn,hcH,hoH,_,hfg,_,hf⟩ :=
not_virtuallyNilpotent_quotient_of_classification_failure Γ h
let _ := hn
exact ⟨H,hn,hcH,hoH,hfg,
quotient_sparse_polynomial_word_growth Γ hc hl ht o d hpoly H hoH,hf⟩
end GraphTopology
end CriticalPercolation
end
end
end OAI
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