diff --git a/Rubin.lean b/Rubin.lean index d3012a9..c597a1e 100644 --- a/Rubin.lean +++ b/Rubin.lean @@ -64,6 +64,292 @@ where end RubinActions +section AlgebraicDisjointness + +variable {G α : Type _} +variable [TopologicalSpace α] +variable [Group G] +variable [ContinuousMulAction G α] +variable [FaithfulSMul G α] + +-- TODO: modify the proof to be less "let everything"-y, especially the first half +lemma proposition_1_1_1 [h_lm : LocallyMoving G α] [T2Space α] (f g : G) (supp_disjoint : Disjoint (Support α f) (Support α g)) : AlgebraicallyDisjoint f g := by + apply AlgebraicallyDisjoint_mk + intros h h_not_commute + -- h is not the identity on `Support α f` + have f_h_not_disjoint := (mt (disjoint_commute (G := G) (α := α)) h_not_commute) + have ⟨x, ⟨x_in_supp_f, x_in_supp_h⟩⟩ := Set.not_disjoint_iff.mp f_h_not_disjoint + have hx_ne_x := mem_support.mp x_in_supp_h + + -- Since α is Hausdoff, there is a nonempty V ⊆ Support α f, with h •'' V disjoint from V + have ⟨V, V_open, x_in_V, V_in_support, disjoint_img_V⟩ := disjoint_nbhd_in (support_open f) x_in_supp_f hx_ne_x + + -- let f₂ be a nontrivial element of the RigidStabilizer G V + let ⟨f₂, f₂_in_rist_V, f₂_ne_one⟩ := h_lm.get_nontrivial_rist_elem V_open (Set.nonempty_of_mem x_in_V) + + -- Re-use the Hausdoff property of α again, this time yielding W ⊆ V + let ⟨y, y_moved⟩ := faithful_moves_point' α f₂_ne_one + have y_in_V := (rigidStabilizer_support.mp f₂_in_rist_V) (mem_support.mpr y_moved) + let ⟨W, W_open, y_in_W, W_in_V, disjoint_img_W⟩ := disjoint_nbhd_in V_open y_in_V y_moved + + -- Let f₁ be a nontrivial element of RigidStabilizer G W + let ⟨f₁, f₁_in_rist_W, f₁_ne_one⟩ := h_lm.get_nontrivial_rist_elem W_open (Set.nonempty_of_mem y_in_W) + + use f₁ + use f₂ + constructor <;> try constructor + · apply disjoint_commute (α := α) + apply Set.disjoint_of_subset_left _ supp_disjoint + calc + Support α f₁ ⊆ W := rigidStabilizer_support.mp f₁_in_rist_W + W ⊆ V := W_in_V + V ⊆ Support α f := V_in_support + · apply disjoint_commute (α := α) + apply Set.disjoint_of_subset_left _ supp_disjoint + calc + Support α f₂ ⊆ V := rigidStabilizer_support.mp f₂_in_rist_V + V ⊆ Support α f := V_in_support + + -- We claim that [f₁, [f₂, h]] is a nontrivial elelement of Centralizer G g + let k := ⁅f₂, h⁆ + have h₂ : ∀ z ∈ W, f₂ • z = k • z := by + intro z z_in_W + simp + symm + apply disjoint_support_comm f₂ h _ disjoint_img_V + · exact W_in_V z_in_W + · exact rigidStabilizer_support.mp f₂_in_rist_V + + constructor + · -- then `k*f₁⁻¹*k⁻¹` is supported on k W = f₂ W, + -- so [f₁,k] is supported on W ∪ f₂ W ⊆ V ⊆ support f, so commutes with g. + apply disjoint_commute (α := α) + apply Set.disjoint_of_subset_left _ supp_disjoint + have supp_f₁_subset_W := (rigidStabilizer_support.mp f₁_in_rist_W) + + show Support α ⁅f₁, ⁅f₂, h⁆⁆ ⊆ Support α f + calc + Support α ⁅f₁, k⁆ = Support α ⁅k, f₁⁆ := by rw [<-commutatorElement_inv, support_inv] + _ ⊆ Support α f₁ ∪ (k •'' Support α f₁) := support_comm α k f₁ + _ ⊆ W ∪ (k •'' Support α f₁) := Set.union_subset_union_left _ supp_f₁_subset_W + _ ⊆ W ∪ (k •'' W) := by + apply Set.union_subset_union_right + exact (smulImage_subset k supp_f₁_subset_W) + _ = W ∪ (f₂ •'' W) := by rw [<-smulImage_eq_of_smul_eq h₂] + _ ⊆ V ∪ (f₂ •'' W) := Set.union_subset_union_left _ W_in_V + _ ⊆ V ∪ V := by + apply Set.union_subset_union_right + apply smulImage_subset_in_support f₂ W V W_in_V + exact rigidStabilizer_support.mp f₂_in_rist_V + _ ⊆ V := by rw [Set.union_self] + _ ⊆ Support α f := V_in_support + + · -- finally, [f₁,k] agrees with f₁ on W, so is not the identity. + have h₄: ∀ z ∈ W, ⁅f₁, k⁆ • z = f₁ • z := by + apply disjoint_support_comm f₁ k + exact rigidStabilizer_support.mp f₁_in_rist_W + rw [<-smulImage_eq_of_smul_eq h₂] + exact disjoint_img_W + let ⟨z, z_in_W, z_moved⟩ := faithful_rigid_stabilizer_moves_point f₁_in_rist_W f₁_ne_one + + by_contra h₅ + rw [<-h₄ z z_in_W] at z_moved + have h₆ : ⁅f₁, ⁅f₂, h⁆⁆ • z = z := by rw [h₅, one_smul] + exact z_moved h₆ +#align proposition_1_1_1 Rubin.proposition_1_1_1 + + +-- TODO: move to Rubin.lean +lemma moves_1234_of_moves_12 {g : G} {x : α} (g12_moves : g^12 • x ≠ x) : + Function.Injective (fun i : Fin 5 => g^(i : ℤ) • x) := +by + apply moves_inj + intros k k_ge_1 k_lt_5 + simp at k_lt_5 + + by_contra x_fixed + have k_div_12 : k ∣ 12 := by + -- Note: norm_num does not support ℤ.dvd yet, nor ℤ.mod, nor Int.natAbs, nor Int.div, etc. + have h: (12 : ℤ) = (12 : ℕ) := by norm_num + rw [h, Int.ofNat_dvd_right] + apply Nat.dvd_of_mod_eq_zero + + interval_cases k + all_goals unfold Int.natAbs + all_goals norm_num + + have g12_fixed : g^12 • x = x := by + rw [<-zpow_ofNat] + simp + rw [<-Int.mul_ediv_cancel' k_div_12] + have res := smul_zpow_eq_of_smul_eq (12/k) x_fixed + group_action at res + exact res + + exact g12_moves g12_fixed + +lemma proposition_1_1_2 [T2Space α] [h_lm : LocallyMoving G α] + (f g : G) (h_disj : AlgebraicallyDisjoint f g) : Disjoint (Support α f) (Support α (g^12)) := +by + by_contra not_disjoint + let U := Support α f ∩ Support α (g^12) + have U_nonempty : U.Nonempty := by + apply Set.not_disjoint_iff_nonempty_inter.mp + exact not_disjoint + + -- Since G is Hausdorff, we can find a nonempty set V ⊆ such that f(V) is disjoint from V and the sets {g^i(V): i=0..4} are pairwise disjoint + let x := U_nonempty.some + have x_in_U : x ∈ U := Set.Nonempty.some_mem U_nonempty + have fx_moves : f • x ≠ x := Set.inter_subset_left _ _ x_in_U + + have five_points : Function.Injective (fun i : Fin 5 => g^(i : ℤ) • x) := by + apply moves_1234_of_moves_12 + exact (Set.inter_subset_right _ _ x_in_U) + have U_open: IsOpen U := (IsOpen.inter (support_open f) (support_open (g^12))) + + let ⟨V₀, V₀_open, x_in_V₀, V₀_in_support, disjoint_img_V₀⟩ := disjoint_nbhd_in U_open x_in_U fx_moves + let ⟨V₁, V₁_open, x_in_V₁, disjoint_img_V₁⟩ := disjoint_nbhd_fin five_points + + let V := V₀ ∩ V₁ + -- Let h be a nontrivial element of the RigidStabilizer G V + let ⟨h, ⟨h_in_ristV, h_ne_one⟩⟩ := h_lm.get_nontrivial_rist_elem (IsOpen.inter V₀_open V₁_open) (Set.nonempty_of_mem ⟨x_in_V₀, x_in_V₁⟩) + + have V_disjoint_smulImage: Disjoint V (f •'' V) := by + apply Set.disjoint_of_subset + · exact Set.inter_subset_left _ _ + · apply smulImage_subset + exact Set.inter_subset_left _ _ + · exact disjoint_img_V₀ + + have comm_non_trivial : ¬Commute f h := by + by_contra comm_trivial + let ⟨z, z_in_V, z_moved⟩ := faithful_rigid_stabilizer_moves_point h_in_ristV h_ne_one + apply z_moved + + nth_rewrite 2 [<-one_smul G z] + rw [<-commutatorElement_eq_one_iff_commute.mpr comm_trivial.symm] + symm + + apply disjoint_support_comm h f + · exact rigidStabilizer_support.mp h_in_ristV + · exact V_disjoint_smulImage + · exact z_in_V + + -- Since g is algebraically disjoint from f, there exist f₁,f₂ ∈ C_G(g) so that the commutator h' = [f1,[f2,h]] is a nontrivial element of C_G(g) + let alg_disj_elem := h_disj h comm_non_trivial + let f₁ := alg_disj_elem.fst + let f₂ := alg_disj_elem.snd + let h' := alg_disj_elem.comm_elem + have f₁_commutes : Commute f₁ g := alg_disj_elem.fst_commute + have f₂_commutes : Commute f₂ g := alg_disj_elem.snd_commute + have h'_commutes : Commute h' g := alg_disj_elem.comm_elem_commute + have h'_nontrivial : h' ≠ 1 := alg_disj_elem.comm_elem_nontrivial + + have support_f₂_h : Support α ⁅f₂,h⁆ ⊆ V ∪ (f₂ •'' V) := by + calc + Support α ⁅f₂, h⁆ ⊆ Support α h ∪ (f₂ •'' Support α h) := support_comm α f₂ h + _ ⊆ V ∪ (f₂ •'' Support α h) := by + apply Set.union_subset_union_left + exact rigidStabilizer_support.mp h_in_ristV + _ ⊆ V ∪ (f₂ •'' V) := by + apply Set.union_subset_union_right + apply smulImage_subset + exact rigidStabilizer_support.mp h_in_ristV + have support_h' : Support α h' ⊆ ⋃(i : Fin 2 × Fin 2), (f₁^(i.1.val) * f₂^(i.2.val)) •'' V := by + rw [rewrite_Union] + simp (config := {zeta := false}) + rw [<-smulImage_mul, <-smulImage_union] + calc + Support α h' ⊆ Support α ⁅f₂,h⁆ ∪ (f₁ •'' Support α ⁅f₂, h⁆) := support_comm α f₁ ⁅f₂,h⁆ + _ ⊆ V ∪ (f₂ •'' V) ∪ (f₁ •'' Support α ⁅f₂, h⁆) := by + apply Set.union_subset_union_left + exact support_f₂_h + _ ⊆ V ∪ (f₂ •'' V) ∪ (f₁ •'' V ∪ (f₂ •'' V)) := by + apply Set.union_subset_union_right + apply smulImage_subset f₁ + exact support_f₂_h + + -- Since h' is nontrivial, it has at least one point p in its support + let ⟨p, p_moves⟩ := faithful_moves_point' α h'_nontrivial + -- Since g commutes with h', all five of the points {gi(p):i=0..4} lie in supp(h') + have gi_in_support : ∀ (i: Fin 5), g^(i.val) • p ∈ Support α h' := by + intro i + rw [mem_support] + by_contra p_fixed + rw [<-mul_smul, h'_commutes.pow_right, mul_smul] at p_fixed + group_action at p_fixed + exact p_moves p_fixed + + -- The next section gets tricky, so let us clear away some stuff first :3 + clear h'_commutes h'_nontrivial + clear V₀_open x_in_V₀ V₀_in_support disjoint_img_V₀ + clear V₁_open x_in_V₁ + clear five_points h_in_ristV h_ne_one V_disjoint_smulImage + clear support_f₂_h + + -- by the pigeonhole principle, one of the four sets V, f₁(V), f₂(V), f₁f₂(V) must contain two of these points, + -- say g^i(p),g^j(p) ∈ k(V) for some 0 ≤ i < j ≤ 4 and k ∈ {1,f₁,f₂,f₁f₂} + let pigeonhole : Fintype.card (Fin 5) > Fintype.card (Fin 2 × Fin 2) := by trivial + let choice_pred := fun (i : Fin 5) => (Set.mem_iUnion.mp (support_h' (gi_in_support i))) + let choice := fun (i : Fin 5) => (choice_pred i).choose + let ⟨i, _, j, _, i_ne_j, same_choice⟩ := Finset.exists_ne_map_eq_of_card_lt_of_maps_to + pigeonhole + (fun (i : Fin 5) _ => Finset.mem_univ (choice i)) + + let k := f₁^(choice i).1.val * f₂^(choice i).2.val + have same_k : f₁^(choice j).1.val * f₂^(choice j).2.val = k := by rw [<-same_choice] + have gi : g^i.val • p ∈ k •'' V := (choice_pred i).choose_spec + have gk : g^j.val • p ∈ k •'' V := by + have gk' := (choice_pred j).choose_spec + rw [same_k] at gk' + exact gk' + + -- Since g^(j-i)(V) is disjoint from V and k commutes with g, + -- we know that g^(j−i)k(V) is disjoint from k(V), + -- which leads to a contradiction since g^i(p) and g^j(p) both lie in k(V). + + have g_disjoint : Disjoint ((g^i.val)⁻¹ •'' V) ((g^j.val)⁻¹ •'' V) := by + apply smulImage_disjoint_subset (Set.inter_subset_right V₀ V₁) + group + rw [smulImage_disjoint_inv_pow] + group + apply disjoint_img_V₁ + symm; exact i_ne_j + + have k_commutes: Commute k g := by + apply Commute.mul_left + · exact f₁_commutes.pow_left _ + · exact f₂_commutes.pow_left _ + clear f₁_commutes f₂_commutes + + have g_k_disjoint : Disjoint ((g^i.val)⁻¹ •'' (k •'' V)) ((g^j.val)⁻¹ •'' (k •'' V)) := by + repeat rw [smulImage_mul] + repeat rw [<-inv_pow] + repeat rw [k_commutes.symm.inv_left.pow_left] + repeat rw [<-smulImage_mul k] + repeat rw [inv_pow] + exact smulImage_disjoint k g_disjoint + + apply Set.disjoint_left.mp g_k_disjoint + · rw [mem_inv_smulImage] + exact gi + · rw [mem_inv_smulImage] + exact gk + +lemma remark_1_2 (f g : G) (h_disj : AlgebraicallyDisjoint f g): Commute f g := by + by_contra non_commute + let disj_elem := h_disj g non_commute + let nontrivial := disj_elem.comm_elem_nontrivial + + rw [commutatorElement_eq_one_iff_commute.mpr disj_elem.snd_commute] at nontrivial + rw [commutatorElement_one_right] at nontrivial + + tauto + +end AlgebraicDisjointness + +section RigidStabilizer + lemma lemma_2_2 (G: Type _) {α : Type _} [Group G] [TopologicalSpace α] [ContinuousMulAction G α] [FaithfulSMul G α] [T2Space α] [h_lm : LocallyMoving G α] {U : Set α} (U_open : IsOpen U) (U_nonempty : Set.Nonempty U) : @@ -196,8 +482,6 @@ by rw [<-period_hg_eq_n] apply Period.pow_period_fix -section proposition_2_1 - -- TODO: put in a different file and introduce some QoL theorems def AlgebraicSubgroup {G : Type _} [Group G] (f : G) : Set G := (fun g : G => g^12) '' { g : G | IsAlgebraicallyDisjoint f g } @@ -263,32 +547,6 @@ by simp at exp_eq_zero exact exp_eq_zero n n_pos -lemma Commute.inv {G : Type _} [Group G] {f g : G} : Commute f g → Commute f g⁻¹ := by - unfold Commute SemiconjBy - intro h - have h₁ : f = g * f * g⁻¹ := by - nth_rw 1 [<-mul_one f] - rw [<-mul_right_inv g, <-mul_assoc] - rw [h] - nth_rw 2 [h₁] - group - -lemma Commute.inv_iff {G : Type _} [Group G] {f g : G} : Commute f g ↔ Commute f g⁻¹ := ⟨ - Commute.inv, - by - nth_rw 2 [<-inv_inv g] - apply Commute.inv -⟩ - -lemma Commute.inv_iff₂ {G : Type _} [Group G] {f g : G} : Commute f g ↔ Commute f⁻¹ g := ⟨ - Commute.symm ∘ Commute.inv_iff.mp ∘ Commute.symm, - Commute.symm ∘ Commute.inv_iff.mpr ∘ Commute.symm -⟩ - -lemma Commute.into {G : Type _} [Group G] {f g : G} : Commute f g → f * g = g * f := by - unfold Commute SemiconjBy - tauto - lemma proposition_2_1 {G α : Type _} [Group G] [TopologicalSpace α] [ContinuousMulAction G α] [T2Space α] [LocallyMoving G α] [h_faithful : FaithfulSMul G α] @@ -373,8 +631,9 @@ by apply h_nc_g12 exact h_in_centralizer _ g12_in_algebraic_subgroup +end RigidStabilizer + -end proposition_2_1 -- variables [topological_space α] [topological_space β] [continuous_mul_action G α] [continuous_mul_action G β] -- noncomputable theorem rubin (hα : rubin_action G α) (hβ : rubin_action G β) : equivariant_homeomorph G α β := sorry end Rubin diff --git a/Rubin/AlgebraicDisjointness.lean b/Rubin/AlgebraicDisjointness.lean index c10819b..df88f1a 100644 --- a/Rubin/AlgebraicDisjointness.lean +++ b/Rubin/AlgebraicDisjointness.lean @@ -239,94 +239,6 @@ by <;> simp only [h, true_or, or_true] · rintro ((h|h)|(h|h)) <;> exact ⟨_, h⟩ --- TODO: move to Rubin.lean --- TODO: modify the proof to be less "let everything"-y, especially the first half --- TODO: use the new class thingy to write a cleaner proof? -lemma proposition_1_1_1 [h_lm : LocallyMoving G α] [T2Space α] (f g : G) (supp_disjoint : Disjoint (Support α f) (Support α g)) : AlgebraicallyDisjoint f g := by - apply AlgebraicallyDisjoint_mk - intros h h_not_commute - -- h is not the identity on `Support α f` - have f_h_not_disjoint := (mt (disjoint_commute (G := G) (α := α)) h_not_commute) - have ⟨x, ⟨x_in_supp_f, x_in_supp_h⟩⟩ := Set.not_disjoint_iff.mp f_h_not_disjoint - have hx_ne_x := mem_support.mp x_in_supp_h - - -- Since α is Hausdoff, there is a nonempty V ⊆ Support α f, with h •'' V disjoint from V - have ⟨V, V_open, x_in_V, V_in_support, disjoint_img_V⟩ := disjoint_nbhd_in (support_open f) x_in_supp_f hx_ne_x - - -- let f₂ be a nontrivial element of the RigidStabilizer G V - let ⟨f₂, f₂_in_rist_V, f₂_ne_one⟩ := h_lm.get_nontrivial_rist_elem V_open (Set.nonempty_of_mem x_in_V) - - -- Re-use the Hausdoff property of α again, this time yielding W ⊆ V - let ⟨y, y_moved⟩ := faithful_moves_point' α f₂_ne_one - have y_in_V := (rigidStabilizer_support.mp f₂_in_rist_V) (mem_support.mpr y_moved) - let ⟨W, W_open, y_in_W, W_in_V, disjoint_img_W⟩ := disjoint_nbhd_in V_open y_in_V y_moved - - -- Let f₁ be a nontrivial element of RigidStabilizer G W - let ⟨f₁, f₁_in_rist_W, f₁_ne_one⟩ := h_lm.get_nontrivial_rist_elem W_open (Set.nonempty_of_mem y_in_W) - - use f₁ - use f₂ - constructor <;> try constructor - · apply disjoint_commute (α := α) - apply Set.disjoint_of_subset_left _ supp_disjoint - calc - Support α f₁ ⊆ W := rigidStabilizer_support.mp f₁_in_rist_W - W ⊆ V := W_in_V - V ⊆ Support α f := V_in_support - · apply disjoint_commute (α := α) - apply Set.disjoint_of_subset_left _ supp_disjoint - calc - Support α f₂ ⊆ V := rigidStabilizer_support.mp f₂_in_rist_V - V ⊆ Support α f := V_in_support - - -- We claim that [f₁, [f₂, h]] is a nontrivial elelement of Centralizer G g - let k := ⁅f₂, h⁆ - have h₂ : ∀ z ∈ W, f₂ • z = k • z := by - intro z z_in_W - simp - symm - apply disjoint_support_comm f₂ h _ disjoint_img_V - · exact W_in_V z_in_W - · exact rigidStabilizer_support.mp f₂_in_rist_V - - constructor - · -- then `k*f₁⁻¹*k⁻¹` is supported on k W = f₂ W, - -- so [f₁,k] is supported on W ∪ f₂ W ⊆ V ⊆ support f, so commutes with g. - apply disjoint_commute (α := α) - apply Set.disjoint_of_subset_left _ supp_disjoint - have supp_f₁_subset_W := (rigidStabilizer_support.mp f₁_in_rist_W) - - show Support α ⁅f₁, ⁅f₂, h⁆⁆ ⊆ Support α f - calc - Support α ⁅f₁, k⁆ = Support α ⁅k, f₁⁆ := by rw [<-commutatorElement_inv, support_inv] - _ ⊆ Support α f₁ ∪ (k •'' Support α f₁) := support_comm α k f₁ - _ ⊆ W ∪ (k •'' Support α f₁) := Set.union_subset_union_left _ supp_f₁_subset_W - _ ⊆ W ∪ (k •'' W) := by - apply Set.union_subset_union_right - exact (smulImage_subset k supp_f₁_subset_W) - _ = W ∪ (f₂ •'' W) := by rw [<-smulImage_eq_of_smul_eq h₂] - _ ⊆ V ∪ (f₂ •'' W) := Set.union_subset_union_left _ W_in_V - _ ⊆ V ∪ V := by - apply Set.union_subset_union_right - apply smulImage_subset_in_support f₂ W V W_in_V - exact rigidStabilizer_support.mp f₂_in_rist_V - _ ⊆ V := by rw [Set.union_self] - _ ⊆ Support α f := V_in_support - - · -- finally, [f₁,k] agrees with f₁ on W, so is not the identity. - have h₄: ∀ z ∈ W, ⁅f₁, k⁆ • z = f₁ • z := by - apply disjoint_support_comm f₁ k - exact rigidStabilizer_support.mp f₁_in_rist_W - rw [<-smulImage_eq_of_smul_eq h₂] - exact disjoint_img_W - let ⟨z, z_in_W, z_moved⟩ := faithful_rigid_stabilizer_moves_point f₁_in_rist_W f₁_ne_one - - by_contra h₅ - rw [<-h₄ z z_in_W] at z_moved - have h₆ : ⁅f₁, ⁅f₂, h⁆⁆ • z = z := by rw [h₅, one_smul] - exact z_moved h₆ -#align proposition_1_1_1 Rubin.proposition_1_1_1 - lemma smul_inj_moves {ι : Type*} [Fintype ι] [T2Space α] {f : ι → G} {x : α} {i j : ι} (i_ne_j : i ≠ j) (f_smul_inj : Function.Injective (fun i : ι => (f i) • x)) : @@ -516,191 +428,4 @@ by exact k_lt_n #align moves_inj_period Rubin.smul_injective_within_period --- TODO: move to Rubin.lean -lemma moves_1234_of_moves_12 {g : G} {x : α} (g12_moves : g^12 • x ≠ x) : - Function.Injective (fun i : Fin 5 => g^(i : ℤ) • x) := -by - apply moves_inj - intros k k_ge_1 k_lt_5 - simp at k_lt_5 - - by_contra x_fixed - have k_div_12 : k ∣ 12 := by - -- Note: norm_num does not support ℤ.dvd yet, nor ℤ.mod, nor Int.natAbs, nor Int.div, etc. - have h: (12 : ℤ) = (12 : ℕ) := by norm_num - rw [h, Int.ofNat_dvd_right] - apply Nat.dvd_of_mod_eq_zero - - interval_cases k - all_goals unfold Int.natAbs - all_goals norm_num - - have g12_fixed : g^12 • x = x := by - rw [<-zpow_ofNat] - simp - rw [<-Int.mul_ediv_cancel' k_div_12] - have res := smul_zpow_eq_of_smul_eq (12/k) x_fixed - group_action at res - exact res - - exact g12_moves g12_fixed - -lemma proposition_1_1_2 [T2Space α] [h_lm : LocallyMoving G α] - (f g : G) (h_disj : AlgebraicallyDisjoint f g) : Disjoint (Support α f) (Support α (g^12)) := -by - by_contra not_disjoint - let U := Support α f ∩ Support α (g^12) - have U_nonempty : U.Nonempty := by - apply Set.not_disjoint_iff_nonempty_inter.mp - exact not_disjoint - - -- Since G is Hausdorff, we can find a nonempty set V ⊆ such that f(V) is disjoint from V and the sets {g^i(V): i=0..4} are pairwise disjoint - let x := U_nonempty.some - have x_in_U : x ∈ U := Set.Nonempty.some_mem U_nonempty - have fx_moves : f • x ≠ x := Set.inter_subset_left _ _ x_in_U - - have five_points : Function.Injective (fun i : Fin 5 => g^(i : ℤ) • x) := by - apply moves_1234_of_moves_12 - exact (Set.inter_subset_right _ _ x_in_U) - have U_open: IsOpen U := (IsOpen.inter (support_open f) (support_open (g^12))) - - let ⟨V₀, V₀_open, x_in_V₀, V₀_in_support, disjoint_img_V₀⟩ := disjoint_nbhd_in U_open x_in_U fx_moves - let ⟨V₁, V₁_open, x_in_V₁, disjoint_img_V₁⟩ := disjoint_nbhd_fin five_points - - let V := V₀ ∩ V₁ - -- Let h be a nontrivial element of the RigidStabilizer G V - let ⟨h, ⟨h_in_ristV, h_ne_one⟩⟩ := h_lm.get_nontrivial_rist_elem (IsOpen.inter V₀_open V₁_open) (Set.nonempty_of_mem ⟨x_in_V₀, x_in_V₁⟩) - - have V_disjoint_smulImage: Disjoint V (f •'' V) := by - apply Set.disjoint_of_subset - · exact Set.inter_subset_left _ _ - · apply smulImage_subset - exact Set.inter_subset_left _ _ - · exact disjoint_img_V₀ - - have comm_non_trivial : ¬Commute f h := by - by_contra comm_trivial - let ⟨z, z_in_V, z_moved⟩ := faithful_rigid_stabilizer_moves_point h_in_ristV h_ne_one - apply z_moved - - nth_rewrite 2 [<-one_smul G z] - rw [<-commutatorElement_eq_one_iff_commute.mpr comm_trivial.symm] - symm - - apply disjoint_support_comm h f - · exact rigidStabilizer_support.mp h_in_ristV - · exact V_disjoint_smulImage - · exact z_in_V - - -- Since g is algebraically disjoint from f, there exist f₁,f₂ ∈ C_G(g) so that the commutator h' = [f1,[f2,h]] is a nontrivial element of C_G(g) - let alg_disj_elem := h_disj h comm_non_trivial - let f₁ := alg_disj_elem.fst - let f₂ := alg_disj_elem.snd - let h' := alg_disj_elem.comm_elem - have f₁_commutes : Commute f₁ g := alg_disj_elem.fst_commute - have f₂_commutes : Commute f₂ g := alg_disj_elem.snd_commute - have h'_commutes : Commute h' g := alg_disj_elem.comm_elem_commute - have h'_nontrivial : h' ≠ 1 := alg_disj_elem.comm_elem_nontrivial - - have support_f₂_h : Support α ⁅f₂,h⁆ ⊆ V ∪ (f₂ •'' V) := by - calc - Support α ⁅f₂, h⁆ ⊆ Support α h ∪ (f₂ •'' Support α h) := support_comm α f₂ h - _ ⊆ V ∪ (f₂ •'' Support α h) := by - apply Set.union_subset_union_left - exact rigidStabilizer_support.mp h_in_ristV - _ ⊆ V ∪ (f₂ •'' V) := by - apply Set.union_subset_union_right - apply smulImage_subset - exact rigidStabilizer_support.mp h_in_ristV - have support_h' : Support α h' ⊆ ⋃(i : Fin 2 × Fin 2), (f₁^(i.1.val) * f₂^(i.2.val)) •'' V := by - rw [rewrite_Union] - simp (config := {zeta := false}) - rw [<-smulImage_mul, <-smulImage_union] - calc - Support α h' ⊆ Support α ⁅f₂,h⁆ ∪ (f₁ •'' Support α ⁅f₂, h⁆) := support_comm α f₁ ⁅f₂,h⁆ - _ ⊆ V ∪ (f₂ •'' V) ∪ (f₁ •'' Support α ⁅f₂, h⁆) := by - apply Set.union_subset_union_left - exact support_f₂_h - _ ⊆ V ∪ (f₂ •'' V) ∪ (f₁ •'' V ∪ (f₂ •'' V)) := by - apply Set.union_subset_union_right - apply smulImage_subset f₁ - exact support_f₂_h - - -- Since h' is nontrivial, it has at least one point p in its support - let ⟨p, p_moves⟩ := faithful_moves_point' α h'_nontrivial - -- Since g commutes with h', all five of the points {gi(p):i=0..4} lie in supp(h') - have gi_in_support : ∀ (i: Fin 5), g^(i.val) • p ∈ Support α h' := by - intro i - rw [mem_support] - by_contra p_fixed - rw [<-mul_smul, h'_commutes.pow_right, mul_smul] at p_fixed - group_action at p_fixed - exact p_moves p_fixed - - -- The next section gets tricky, so let us clear away some stuff first :3 - clear h'_commutes h'_nontrivial - clear V₀_open x_in_V₀ V₀_in_support disjoint_img_V₀ - clear V₁_open x_in_V₁ - clear five_points h_in_ristV h_ne_one V_disjoint_smulImage - clear support_f₂_h - - -- by the pigeonhole principle, one of the four sets V, f₁(V), f₂(V), f₁f₂(V) must contain two of these points, - -- say g^i(p),g^j(p) ∈ k(V) for some 0 ≤ i < j ≤ 4 and k ∈ {1,f₁,f₂,f₁f₂} - let pigeonhole : Fintype.card (Fin 5) > Fintype.card (Fin 2 × Fin 2) := by trivial - let choice_pred := fun (i : Fin 5) => (Set.mem_iUnion.mp (support_h' (gi_in_support i))) - let choice := fun (i : Fin 5) => (choice_pred i).choose - let ⟨i, _, j, _, i_ne_j, same_choice⟩ := Finset.exists_ne_map_eq_of_card_lt_of_maps_to - pigeonhole - (fun (i : Fin 5) _ => Finset.mem_univ (choice i)) - - let k := f₁^(choice i).1.val * f₂^(choice i).2.val - have same_k : f₁^(choice j).1.val * f₂^(choice j).2.val = k := by rw [<-same_choice] - have gi : g^i.val • p ∈ k •'' V := (choice_pred i).choose_spec - have gk : g^j.val • p ∈ k •'' V := by - have gk' := (choice_pred j).choose_spec - rw [same_k] at gk' - exact gk' - - -- Since g^(j-i)(V) is disjoint from V and k commutes with g, - -- we know that g^(j−i)k(V) is disjoint from k(V), - -- which leads to a contradiction since g^i(p) and g^j(p) both lie in k(V). - - have g_disjoint : Disjoint ((g^i.val)⁻¹ •'' V) ((g^j.val)⁻¹ •'' V) := by - apply smulImage_disjoint_subset (Set.inter_subset_right V₀ V₁) - group - rw [smulImage_disjoint_inv_pow] - group - apply disjoint_img_V₁ - symm; exact i_ne_j - - have k_commutes: Commute k g := by - apply Commute.mul_left - · exact f₁_commutes.pow_left _ - · exact f₂_commutes.pow_left _ - clear f₁_commutes f₂_commutes - - have g_k_disjoint : Disjoint ((g^i.val)⁻¹ •'' (k •'' V)) ((g^j.val)⁻¹ •'' (k •'' V)) := by - repeat rw [smulImage_mul] - repeat rw [<-inv_pow] - repeat rw [k_commutes.symm.inv_left.pow_left] - repeat rw [<-smulImage_mul k] - repeat rw [inv_pow] - exact smulImage_disjoint k g_disjoint - - apply Set.disjoint_left.mp g_k_disjoint - · rw [mem_inv_smulImage] - exact gi - · rw [mem_inv_smulImage] - exact gk - -lemma remark_1_2 (f g : G) (h_disj : AlgebraicallyDisjoint f g): Commute f g := by - by_contra non_commute - let disj_elem := h_disj g non_commute - let nontrivial := disj_elem.comm_elem_nontrivial - - rw [commutatorElement_eq_one_iff_commute.mpr disj_elem.snd_commute] at nontrivial - rw [commutatorElement_one_right] at nontrivial - - tauto - end Rubin