| import Mathlib |
| import OAI.Probability.CriticalPercolation.Growth.CosetPresentation |
| import OAI.Probability.CriticalPercolation.Harmonic.WeakGromovHarmonic |
| import OAI.Probability.CriticalPercolation.Growth.CosetImageGrowth |
| import OAI.Probability.CriticalPercolation.Harmonic.GraphHarmonicLinear |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set Function |
| open scoped Classical Pointwise |
|
|
| namespace CriticalPercolation.RelativeGrowth |
| variable {A B : Type*} [Group A] [Group B] |
|
|
| lemma finite_image_to_integer_ker (K : Subgroup A) (f : A →* Multiplicative ℤ) |
| (hf : (f '' (K:Set A)).Finite) : K≤f.ker := by |
| let J := K.map f |
| let : Finite J := hf.to_subtype |
| intro k hk |
| let j : J := ⟨f k,Subgroup.mem_map.mpr ⟨k,hk,rfl⟩⟩ |
| obtain ⟨n,hn,he⟩ := isOfFinOrder_iff_pow_eq_one.mp (isOfFinOrder_of_finite j) |
| have hm : (f k)^n=1 := congrArg Subtype.val he |
| exact (isOfFinOrder_iff_pow_eq_one.mpr ⟨n,hn,hm⟩).eq_one' |
|
|
| lemma indicability_pullback_killing (K : Subgroup A) (f : A →* B) (hf : Surjective f) |
| (hK : (f '' (K:Set A)).Finite) |
| (hb : ∃ P : Subgroup B, P.FiniteIndex ∧ ∃ q : P →* Multiplicative ℤ, Surjective q) : |
| ∃ H : Subgroup A, H.FiniteIndex ∧ ∃ q : H →* Multiplicative ℤ, |
| Surjective q ∧ K.subgroupOf H≤q.ker := by |
| obtain ⟨P,hPi,q,hq⟩ := hb |
| let := hPi |
| let H := P.comap f |
| let ψ : H →* P := (f.comp H.subtype).codRestrict P (fun h => h.property) |
| have hψ : Surjective ψ := by |
| intro p |
| obtain ⟨a,ha⟩ := hf p.val |
| refine ⟨⟨a,?_⟩,Subtype.ext ha⟩ |
| change f a∈P |
| rw [ha] |
| exact p.property |
| refine ⟨H,LinearGrowth.finiteIndex_comap_surjective f hf P,q.comp ψ,hq.comp hψ,?_⟩ |
| apply finite_image_to_integer_ker |
| change ((fun h => q (ψ h)) '' (K.subgroupOf H:Set H)).Finite |
| rw [←Set.image_image q ψ] |
| apply Set.Finite.image |
| apply Set.Finite.of_finite_image (f:=P.subtype) |
| · apply hK.subset |
| rintro _ ⟨p,⟨h,hh,rfl⟩,rfl⟩ |
| exact ⟨h.val,hh,rfl⟩ |
| · exact P.subtype_injective.injOn |
|
|
| lemma Presentation.virtually_indicable (P : Presentation A) (d : ℕ) |
| (hg : HasDegree P.K P.S d) [Infinite P.X] : |
| ∃ H : Subgroup A, H.FiniteIndex ∧ ∃ q : H →* Multiplicative ℤ, |
| Surjective q ∧ P.K.subgroupOf H≤q.ker := by |
| let W := P.geometry (hg.1 1) |
| let ρ := (W.harmonicRepresentation 1).comp P.actionHom |
| have hWg : W.HasDegree d := P.geometry_hasDegree d hg |
| have htrans : ∀ x : P.X, ∃ a : A, P.actionHom a W.root=x := by |
| intro x |
| obtain ⟨a,rfl⟩ := QuotientGroup.mk_surjective x |
| exact ⟨a,by change P.projection (a*1)=P.projection a; rw [mul_one]⟩ |
| have hInf : (Set.range ρ).Infinite := W.infinite_harmonic_linear_image P.actionHom d hWg htrans |
| have hKρ : (ρ '' (P.K:Set A)).Finite := by |
| apply (W.finite_stabilizer_harmonic_image d 1 hWg).subset |
| rintro _ ⟨k,hk,rfl⟩ |
| have hstab : P.actionHom k∈W.Stab := by |
| change P.projection (k*1)=P.projection 1 |
| apply QuotientGroup.eq.mpr |
| simpa using P.K.inv_mem hk |
| exact ⟨⟨P.actionHom k,hstab⟩,rfl⟩ |
| let f := ρ.rangeRestrict |
| have hKf : (f '' (P.K:Set A)).Finite := by |
| apply Set.Finite.of_finite_image (f:=ρ.range.subtype) |
| · simpa only [Set.image_image] using (show ((fun x => ρ.range.subtype (f x)) '' (P.K:Set A)).Finite from hKρ) |
| · exact ρ.range.subtype_injective.injOn |
| have hgR := hg.image_polynomial P.one_mem P.inv_eq P.generates f ρ.rangeRestrict_surjective hKf |
| let : Infinite ρ.range := hInf.to_subtype |
| exact indicability_pullback_killing P.K f ρ.rangeRestrict_surjective hKf hgR.linear_virtually_indicable |
|
|
| lemma virtually_indicable {K : Subgroup A} {S : Set A} {d : ℕ} |
| (h1 : (1:A)∈S) (hi : S⁻¹=S) (hgen : Subgroup.closure S=⊤) |
| (hK : Bounded S (K:Set A)) (hg : HasDegree K S d) [Infinite (A ⧸ K)] : |
| ∃ H : Subgroup A, H.FiniteIndex ∧ ∃ q : H →* Multiplicative ℤ, |
| Surjective q ∧ K.subgroupOf H≤q.ker := by |
| obtain ⟨P,hPK,hPb,hSP⟩ := normalized_generators K S h1 hi hgen hK |
| have hgP : HasDegree P.K P.S d := hPK.symm ▸ hg.bounded_generators h1 hPb |
| let : Infinite P.X := by change Infinite (A ⧸ P.K); rw [hPK]; infer_instance |
| exact hPK ▸ P.virtually_indicable d hgP |
| |
| end CriticalPercolation.RelativeGrowth |
| end |
| end |
| |
| end OAI |
| |
