File size: 3,322 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 71 72 73 74 75 76 77 78 79 80 81 82 | import Mathlib
import OAI.Probability.CriticalPercolation.Criticality.VirtualIndicability
import OAI.Probability.CriticalPercolation.Nilpotent.FiniteKernelNilpotent
import OAI.Probability.CriticalPercolation.Growth.KernelGrowthDescent
import OAI.Probability.CriticalPercolation.LinearGroups.SolvablePeriodicCenter
namespace OAI
section
noncomputable section
open Set Function
open scoped Classical IsMulCommutative
namespace CriticalPercolation.SparseGrowth
universe u
theorem solvable_sparse_polynomial_virtuallyNilpotent (d : ℕ) :
∀ (A : Type u) [Group A] [DecidableEq A] [Group.IsSolvable A],
HasPolynomialDegree A d → Group.IsVirtuallyNilpotent A := by
induction d with
| zero =>
intro A _ _ _ hg
let _ : Finite A := hg.finite
exact virtuallyNilpotent_of_finite
| succ d ih =>
intro A _ _ _ hg
rcases finite_or_infinite A with hfin | hinf
· let _ := hfin
exact virtuallyNilpotent_of_finite
let _ := hinf
obtain ⟨H,hHi,f,hf⟩ := hg.virtually_indicable
let _ := hHi
apply virtuallyNilpotent_of_finiteIndex H
have hgH := hg.finiteIndex_subgroup H
let _ : Group.FG H := hgH.fg
let K := f.ker
let _ : Group.FG K := hgH.solvable_subgroup_fg K
let _ : Infinite (H ⧸ K) :=
(QuotientGroup.quotientKerEquivOfSurjective f hf).toEquiv.infinite_iff.mpr inferInstance
have hgK := hgH.normal_subgroup_descent K
have hvK := ih K hgK
obtain ⟨P,hP,hPi,hPn⟩ := hgK.characteristic_nilpotent_core hvK
let _ := hP
let _ := hPi
let _ := hPn
let L := P.map K.subtype
let e : P ≃* L := P.equivMapOfInjective K.subtype Subtype.val_injective
let _ : L.Normal := characteristic_map_normal K P
let _ : Group.IsNilpotent L := Group.nilpotent_of_surjective e.toMonoidHom e.surjective
have hLK : L≤K := Subgroup.map_subtype_le P
have he : L.subgroupOf K=P := by
exact Subgroup.comap_map_eq_self_of_injective K.subtype_injective P
let _ : (L.subgroupOf K).FiniteIndex := he.symm ▸ hPi
rcases finite_or_infinite L with hLf | hLi
· let _ := hLf
let _ : Finite P := Finite.of_equiv L e.symm.toEquiv
let _ : Finite K := P.finite_iff_finite_and_finiteIndex.mpr ⟨inferInstance,hPi⟩
exact CompactQuotient.virtuallyNilpotent_of_finite_kernel f (Set.toFinite _)
(Group.IsNilpotent.isVirtuallyNilpotent inferInstance)
· let _ := hLi
obtain ⟨J,hJi,x,hx,hxinf⟩ := hgH.periodic_center_of_nilpotent_kernel f hf L hLK
let _ := hJi
apply virtuallyNilpotent_of_finiteIndex J
have hgJ := hgH.finiteIndex_subgroup J
have hgQ := hgJ.quotient_descent (Subgroup.center J) x hx (orderOf_eq_zero hxinf)
have hvQ := ih (J ⧸ Subgroup.center J) hgQ
exact virtuallyNilpotent_of_central_kernel (QuotientGroup.mk' (Subgroup.center J))
(by rw [QuotientGroup.ker_mk']) hvQ
theorem HasPolynomialDegree.virtuallyNilpotent_of_solvable_finiteIndex
{A : Type u} [Group A] [DecidableEq A] {d : ℕ}
(hg : HasPolynomialDegree A d) (H : Subgroup A) [H.FiniteIndex]
[Group.IsSolvable H] : Group.IsVirtuallyNilpotent A := by
exact virtuallyNilpotent_of_finiteIndex H
(solvable_sparse_polynomial_virtuallyNilpotent d H (hg.finiteIndex_subgroup H))
end CriticalPercolation.SparseGrowth
end
end
end OAI
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