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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Classification.RelativeClassification
import OAI.Probability.CriticalPercolation.Growth.GraphRelativeGrowth
import OAI.Probability.CriticalPercolation.GraphTopology.CommensurableClosure
import OAI.Probability.CriticalPercolation.Classification.DiscreteWeakGromovBridge
import OAI.Probability.CriticalPercolation.GraphGeometry.GraphQuotientPolynomial
namespace OAI
section
noncomputable section
open Set Function Topology
open scoped Classical Pointwise
namespace CriticalPercolation.GraphTopology
variable {X : Type*} (Γ : SimpleGraph X)
theorem compact_open_normal_of_sparse_polynomial
[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) :
let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ
∃ H : Subgroup (Γ ≃g Γ), H.Normal ∧ IsCompact (H:Set (Γ ≃g Γ)) ∧ IsOpen (H:Set (Γ ≃g Γ)) := by
let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ
let K := vertexStabilizer Γ o
let S := displacement Γ o 1
have h1 : (1:Γ ≃g Γ)∈S := one_mem_displacement Γ o 1
have hi : S⁻¹=S := by ext g; exact displacement_inv Γ
have hgen : Subgroup.closure S=⊤ := displacement_generates Γ hc ht o
obtain ⟨N,hNn,hNb,hNi⟩ := RelativeGrowth.bounded_normal_commensurable d
(Γ ≃g Γ) K S h1 hi hgen (vertexStabilizer_bounded Γ o) (displacement_relativeDegree Γ hl o d hpoly)
let := hNn
let H := N.topologicalClosure
have hHn : H.Normal := Subgroup.is_normal_topologicalClosure N
have hcH : IsCompact (H:Set (Γ ≃g Γ)) := by
obtain ⟨n,hn⟩ := hNb
have hNs : (N:Set (Γ ≃g Γ))⊆displacement Γ o n := hn.trans (displacement_pow_subset Γ o n)
apply (compact_displacement Γ hc hl o n).of_isClosed_subset N.isClosed_topologicalClosure
exact closure_minimal hNs (isClosed_eval_mem Γ o _)
have hoH : IsOpen (H:Set (Γ ≃g Γ)) :=
RelativeGrowth.topologicalClosure_open_of_relIndex K N (open_stabilizer Γ o) hNi
exact ⟨H,hHn,hcH,hoH⟩
theorem compact_open_virtually_nilpotent_quotient_of_sparse_polynomial
(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) :
HasCompactOpenVirtuallyNilpotentQuotient Γ := by
let _ : TopologicalSpace X := ⊥
let _ : DiscreteTopology X := ⟨rfl⟩
let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ
obtain ⟨H,hHn,hcH,hoH⟩ := compact_open_normal_of_sparse_polynomial Γ hc hl ht o d hpoly
let := hHn
have hg := quotient_sparse_polynomial_word_growth Γ hc hl ht o d hpoly H hoH
obtain ⟨N,hNn,hNi⟩ := CompactQuotient.virtuallyNilpotent_of_sparse_word_growth hg
exact ⟨H,hHn,hcH,hoH,N,hNn,hNi.index_ne_zero⟩
end CriticalPercolation.GraphTopology
end
end
end OAI
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