| import Mathlib |
|
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| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set Function |
| open scoped Pointwise |
|
|
| namespace CriticalPercolation.CosetGrowth |
| variable {A : Type*} [Group A] |
|
|
| def ball (S : Set A) (L : Subgroup A) (n : ℕ) : Set (A ⧸ L) := |
| (QuotientGroup.mk : A → A ⧸ L) '' (S^n) |
|
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| lemma ball_zero (S : Set A) (L : Subgroup A) : ball S L 0={QuotientGroup.mk (1 : A)} := by |
| simp [ball] |
|
|
| lemma ball_step (S : Set A) (L : Subgroup A) (n : ℕ) : |
| ball S L (n+1)=⋃ s∈S, (fun q : A ⧸ L => s • q) '' ball S L n := by |
| ext q |
| simp only [ball,pow_succ',Set.mem_image,Set.mem_mul,Set.mem_iUnion] |
| constructor |
| · rintro ⟨_,⟨s,hs,a,ha,rfl⟩,rfl⟩ |
| exact ⟨s,hs,QuotientGroup.mk a,⟨a,ha,rfl⟩,rfl⟩ |
| · rintro ⟨s,hs,_,⟨a,ha,rfl⟩,rfl⟩ |
| exact ⟨s*a,⟨s,hs,a,ha,rfl⟩,rfl⟩ |
|
|
| lemma ball_mono (S : Set A) (L : Subgroup A) (h1 : (1 : A)∈S) : |
| Monotone (ball S L) := fun _ _ h => Set.image_mono (Set.pow_subset_pow_right h1 h) |
|
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| lemma ball_one_mem (S : Set A) (L : Subgroup A) (h1 : (1 : A)∈S) (n : ℕ) : |
| QuotientGroup.mk (1 : A)∈ball S L n := by |
| apply ball_mono S L h1 (Nat.zero_le n) |
| rw [ball_zero] |
| exact Set.mem_singleton _ |
|
|
| lemma ball_eq_univ_of_stationary (S : Set A) (L : Subgroup A) |
| (h1 : (1 : A)∈S) (hinv : S⁻¹=S) (hgen : Subgroup.closure S=⊤) |
| (n : ℕ) (hstop : ball S L (n+1)=ball S L n) : ball S L n=Set.univ := by |
| have hstable : ∀ s∈S, ∀ q∈ball S L n, s • q∈ball S L n := by |
| intro s hs q hq |
| rw [←hstop,ball_step] |
| exact Set.mem_iUnion.mpr ⟨s,Set.mem_iUnion.mpr ⟨hs,⟨q,hq,rfl⟩⟩⟩ |
| have hS : S⊆MulAction.stabilizer A (ball S L n) := by |
| intro s hs |
| change (fun q : A ⧸ L => s • q) '' ball S L n=ball S L n |
| apply Set.Subset.antisymm |
| · rintro _ ⟨q,hq,rfl⟩ |
| exact hstable s hs q hq |
| · intro q hq |
| have hi : s⁻¹∈S := by rw [←hinv]; simpa only [Set.mem_inv,inv_inv] using hs |
| exact ⟨s⁻¹ • q,hstable _ hi _ hq,smul_inv_smul s q⟩ |
| have hall : ∀ g : A, g • ball S L n=ball S L n := by |
| have hh := (Subgroup.closure_le (MulAction.stabilizer A (ball S L n))).mpr hS |
| rw [hgen] at hh |
| exact fun g => hh (Subgroup.mem_top g) |
| apply Set.eq_univ_of_forall |
| intro q |
| obtain ⟨g,rfl⟩ := QuotientGroup.mk_surjective q |
| have hh : g • (QuotientGroup.mk (1 : A) : A ⧸ L)∈ball S L n := by |
| rw [←hall g] |
| exact ⟨_,ball_one_mem S L h1 n,rfl⟩ |
| simpa only [MulAction.Quotient.smul_mk,smul_eq_mul,mul_one] using hh |
|
|
| lemma finiteIndex_of_small_ball (S : Set A) (L : Subgroup A) |
| (h1 : (1 : A)∈S) (hinv : S⁻¹=S) (hgen : Subgroup.closure S=⊤) |
| (n : ℕ) (hfin : (ball S L n).Finite) (hsmall : (ball S L n).ncard≤n) : |
| L.FiniteIndex := by |
| have hmon := ball_mono S L h1 |
| have hfi : ∀ i≤n, (ball S L i).Finite := fun i hi => hfin.subset (hmon hi) |
| have hs : ∃ i<n, ball S L (i+1)=ball S L i := by |
| by_contra! h |
| have hg : ∀ i≤n, i+1≤(ball S L i).ncard := by |
| intro i hi |
| induction i with |
| | zero => simp [ball_zero] |
| | succ i ih => |
| have hle := ih (by omega) |
| have hlt : (ball S L i).ncard<(ball S L (i+1)).ncard := by |
| apply Set.ncard_lt_ncard |
| · exact Set.ssubset_iff_subset_ne.mpr ⟨hmon (Nat.le_succ i),Ne.symm (h i (by omega))⟩ |
| · exact hfi _ hi |
| omega |
| have := hg n le_rfl |
| omega |
| obtain ⟨i,hi,he⟩ := hs |
| have heq := ball_eq_univ_of_stationary S L h1 hinv hgen i he |
| have hf : (Set.univ : Set (A ⧸ L)).Finite := heq ▸ hfi i (by omega) |
| let _ : Finite (A ⧸ L) := Set.finite_univ_iff.mp hf |
| exact Subgroup.finiteIndex_of_finite_quotient |
|
|
| lemma ball_subset_coset_representatives (S : Set A) (L : Subgroup A) |
| (n : ℕ) (F : Finset A) (hcover : S^n⊆(F : Set A)*(L : Set A)) : |
| ball S L n⊆(QuotientGroup.mk : A → A ⧸ L) '' (F : Set A) := by |
| rintro q ⟨a,ha,rfl⟩ |
| obtain ⟨g,hg,l,hl,rfl⟩ := hcover ha |
| refine ⟨g,hg,?_⟩ |
| apply (@QuotientGroup.eq A _ L g (g*l)).mpr |
| rw [inv_mul_cancel_left] |
| exact hl |
|
|
| theorem finiteIndex_of_few_cosets (S : Set A) (L : Subgroup A) |
| (h1 : (1 : A)∈S) (hinv : S⁻¹=S) (hgen : Subgroup.closure S=⊤) |
| (n : ℕ) (F : Finset A) (hF : F.card≤n) |
| (hcover : S^n⊆(F : Set A)*(L : Set A)) : L.FiniteIndex := by |
| have hi := ball_subset_coset_representatives S L n F hcover |
| have hfin := (F.finite_toSet.image (QuotientGroup.mk : A → A ⧸ L)).subset hi |
| apply finiteIndex_of_small_ball S L h1 hinv hgen n hfin |
| apply (Set.ncard_le_ncard hi (F.finite_toSet.image _)).trans |
| apply (Set.ncard_image_le F.finite_toSet).trans |
| simpa using hF |
|
|
| end CriticalPercolation.CosetGrowth |
| end |
| end |
|
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| end OAI |
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