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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Classification.WeakGromovMainBoundary
import OAI.Probability.CriticalPercolation.Harmonic.WeakGromovHarmonic
namespace OAI
section
noncomputable section
open Set Function Topology
open scoped Classical Pointwise
namespace CriticalPercolation.CompactQuotient
lemma virtuallyNilpotent_of_sparse_word_growth {A : Type*} [Group A]
(hg : HasSparsePolynomialWordGrowth A) : Group.IsVirtuallyNilpotent A := by
obtain ⟨S,hSf,h1,hi,hgen,d,hg⟩ := hg
exact SparseGrowth.sparse_polynomial_virtuallyNilpotent d A
⟨S,hSf,h1,hi,hgen,1,by simpa only [one_mul] using hg⟩
end CriticalPercolation.CompactQuotient
namespace CriticalPercolation.GraphTopology
lemma no_weak_Gromov_quotient_counterexample {X : Type*} (Γ : SimpleGraph X) :
¬HasWeakGromovQuotientCounterexample Γ := by
let _ : TopologicalSpace X := ⊥
let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ
rintro ⟨H,hn,_,_,_,hg,hfail⟩
let _ := hn
exact hfail (CompactQuotient.virtuallyNilpotent_of_sparse_word_growth hg)
end CriticalPercolation.GraphTopology
end
end
end OAI
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