openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Criticality /IndicableDegreeStep.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Nilpotent.FiniteKernelNilpotent | |
| import OAI.Probability.CriticalPercolation.Growth.KernelGrowthDescent | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.IntegerKernelGeneration | |
| import OAI.Probability.CriticalPercolation.Nilpotent.GeneralNilpotentCore | |
| import OAI.Probability.CriticalPercolation.LinearGroups.GeneralPeriodicCenter | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function | |
| open scoped Classical IsMulCommutative | |
| namespace CriticalPercolation.SparseGrowth | |
| universe u | |
| theorem virtuallyNilpotent_of_indicability_degree_step | |
| (d : ℕ) | |
| (ih : ∀ (B : Type u) [Group B] [DecidableEq B], | |
| HasPolynomialDegree B d → Group.IsVirtuallyNilpotent B) | |
| (A : Type u) [Group A] [DecidableEq A] | |
| (hg : HasPolynomialDegree A (d+1)) | |
| (hind : ∃ H : Subgroup A, H.FiniteIndex ∧ | |
| ∃ f : H →* Multiplicative ℤ, Surjective f) : | |
| Group.IsVirtuallyNilpotent A := by | |
| rcases finite_or_infinite A with hfin | hinf | |
| · let _ := hfin | |
| exact virtuallyNilpotent_of_finite | |
| let _ := hinf | |
| obtain ⟨H,hHi,f,hf⟩ := hind | |
| 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.integer_kernel_fg f hf | |
| 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_general 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 | |
| let _ : Group.FG P := inferInstance | |
| let _ : Group.FG L := Group.fg_of_surjective (f := e.toMonoidHom) e.surjective | |
| obtain ⟨J,hJi,x,hx,hxinf⟩ := hgH.periodic_center_of_fg_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 | |
| end CriticalPercolation.SparseGrowth | |
| end | |
| end | |
| end OAI | |