File size: 2,680 Bytes
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 | import Mathlib
import OAI.Probability.CriticalPercolation.Classification.FiniteClassificationObstruction
import OAI.Probability.CriticalPercolation.Nilpotent.FiniteKernelNilpotent
namespace OAI
section
noncomputable section
open Set Function Topology
namespace CriticalPercolation.CompactQuotient
variable {A : Type*} [Group A]
lemma virtuallyNilpotent_of_finiteByVirtuallyNilpotent
(h : FiniteByVirtuallyNilpotent A) : Group.IsVirtuallyNilpotent A := by
obtain ⟨F,hn,hF,N,hN,hi⟩ := h
let _ := hn
apply virtuallyNilpotent_of_finite_kernel (QuotientGroup.mk' F)
· simpa only [QuotientGroup.ker_mk'] using hF
· exact ⟨N,hN,⟨hi⟩⟩
lemma finiteByVirtuallyNilpotent_of_virtuallyNilpotent
(h : Group.IsVirtuallyNilpotent A) : FiniteByVirtuallyNilpotent A := by
obtain ⟨N,hN,hi⟩ := h
let _ := hN
let f : A →* A ⧸ (⊥ : Subgroup A) := QuotientGroup.mk' ⊥
have hf : Surjective f := QuotientGroup.mk_surjective
have hnil : Group.IsNilpotent (N.map f) := Group.nilpotent_of_surjective
(f.subgroupMap N) (f.subgroupMap_surjective N)
have hindex : (N.map f).index=N.index := by
apply N.index_map_eq hf
change (QuotientGroup.mk' (⊥ : Subgroup A)).ker≤N
rw [QuotientGroup.ker_mk']
exact bot_le
refine ⟨⊥,inferInstance,by simp,N.map f,hnil,?_⟩
rw [hindex]
exact hi.index_ne_zero
theorem finiteByVirtuallyNilpotent_iff :
FiniteByVirtuallyNilpotent A ↔ Group.IsVirtuallyNilpotent A :=
⟨virtuallyNilpotent_of_finiteByVirtuallyNilpotent,
finiteByVirtuallyNilpotent_of_virtuallyNilpotent⟩
end CriticalPercolation.CompactQuotient
namespace CriticalPercolation.GraphTopology
variable {X : Type*} (Γ : SimpleGraph X)
theorem not_virtuallyNilpotent_quotient_of_classification_failure {K : ℝ}
(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)) ∧
1≤K ∧ Group.FG ((Γ ≃g Γ) ⧸ H) ∧
CompactQuotient.UniformFiniteApproximatePowers ((Γ ≃g Γ) ⧸ H) K ∧
¬Group.IsVirtuallyNilpotent ((Γ ≃g Γ) ⧸ H) := by
let _ : TopologicalSpace X := ⊥
let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ
obtain ⟨H,hn,hc,ho,hfg,hpow,hfail⟩ := h
let _ := hn
exact ⟨H,hn,hc,ho,CompactQuotient.one_le_of_uniformFiniteApproximatePowers hpow,
hfg,hpow,fun hv => hfail
(CompactQuotient.finiteByVirtuallyNilpotent_of_virtuallyNilpotent hv)⟩
end CriticalPercolation.GraphTopology
end
end
end OAI
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