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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Nilpotent.NilpotentSubgroupFG
import OAI.Probability.CriticalPercolation.Nilpotent.AbelianConjugationGrowth
import OAI.Probability.CriticalPercolation.Nilpotent.NilpotentInfiniteCenter
import OAI.Probability.CriticalPercolation.Nilpotent.FiniteCyclicExtensionIndex
namespace OAI
section
noncomputable section
open Set Function
open scoped Classical IsMulCommutative
namespace CriticalPercolation.SparseGrowth
variable {A : Type*} [Group A] [DecidableEq A]
lemma HasPolynomialDegree.periodic_center_of_fg_nilpotent_kernel {d : ℕ}
(hg : HasPolynomialDegree A d)
(f : A →* Multiplicative ℤ) (hf : Surjective f)
(L : Subgroup A) [L.Normal] [Group.IsNilpotent L] [Group.FG L] [Infinite L]
(hLK : L ≤ f.ker) [(L.subgroupOf f.ker).FiniteIndex] :
∃ H : Subgroup A, H.FiniteIndex ∧ ∃ x : H,
x∈Subgroup.center H ∧ ¬IsOfFinOrder x := by
let Z := Subgroup.center L
let _ : Infinite Z := infinite_center_of_nilpotent
let C := Z.map L.subtype
let e : Z ≃* C := Z.equivMapOfInjective L.subtype Subtype.val_injective
let _ : C.Normal := characteristic_map_normal L Z
let _ : IsMulCommutative C := ⟨⟨fun x y => e.symm.injective (by
simp only [map_mul]; exact mul_comm _ _)⟩⟩
let _ : Infinite C := Infinite.of_injective e e.injective
let _ : Group.FG Z := nilpotent_subgroup_fg L Z
let _ : Group.FG C := Group.fg_of_surjective (f := e.toMonoidHom) e.surjective
obtain ⟨a,ha⟩ := hf (Multiplicative.ofAdd 1)
obtain ⟨m,hm,x,hx,hfix⟩ := hg.abelian_normal_periodic_direction C a
let H := Subgroup.centralizer ({x.val} : Set A)
have hLH : L≤H := by
intro l hl
obtain ⟨z,hz,hzx⟩ := x.property
rw [Subgroup.mem_centralizer_singleton_iff,←hzx]
exact congrArg Subtype.val ((Subgroup.mem_center_iff.mp hz) ⟨l,hl⟩)
have haH : a^m∈H := by
rw [Subgroup.mem_centralizer_singleton_iff]
have he := congrArg Subtype.val hfix
change a^m*x.val*(a^m)⁻¹=x.val at he
exact mul_inv_eq_iff_eq_mul.mp he
have hHi : H.FiniteIndex := finiteIndex_of_normal_kernel_subgroup f L hLK a ha H hLH m hm haH
let y : H := ⟨x.val,by rw [Subgroup.mem_centralizer_singleton_iff]⟩
refine ⟨H,hHi,y,?_,?_⟩
· rw [Subgroup.mem_center_iff]
intro w
apply Subtype.ext
exact Subgroup.mem_centralizer_singleton_iff.mp w.property
· intro hy
obtain ⟨k,hk,he⟩ := hy.exists_pow_eq_one
apply hx
apply isOfFinOrder_iff_pow_eq_one.mpr
exact ⟨k,hk,Subtype.ext (congrArg (fun z : H => z.val) he)⟩
end CriticalPercolation.SparseGrowth
end
end
end OAI
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