/- Copyright (c) 2023 Laurent Bartholdi. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Author : Laurent Bartholdi -/ import Mathlib.Data.Finset.Basic import Mathlib.Data.Finset.Card import Mathlib.Data.Fintype.Perm import Mathlib.GroupTheory.Subgroup.Basic import Mathlib.GroupTheory.Commutator import Mathlib.GroupTheory.GroupAction.Basic import Mathlib.GroupTheory.Exponent import Mathlib.GroupTheory.Perm.Basic import Mathlib.Topology.Basic import Mathlib.Topology.Bases import Mathlib.Topology.Compactness.Compact import Mathlib.Topology.Separation import Mathlib.Topology.Homeomorph import Rubin.Tactic import Rubin.MulActionExt import Rubin.SmulImage import Rubin.Support import Rubin.Topology import Rubin.RigidStabilizer import Rubin.RigidStabilizerBasis import Rubin.Period import Rubin.AlgebraicDisjointness import Rubin.RegularSupport import Rubin.RegularSupportBasis import Rubin.HomeoGroup #align_import rubin namespace Rubin open Rubin.Tactic -- TODO: find a home theorem equiv_congr_ne {ι ι' : Type _} (e : ι ≃ ι') {x y : ι} : x ≠ y → e x ≠ e y := by intro x_ne_y by_contra h apply x_ne_y convert congr_arg e.symm h <;> simp only [Equiv.symm_apply_apply] #align equiv.congr_ne Rubin.equiv_congr_ne ---------------------------------------------------------------- section Rubin ---------------------------------------------------------------- section RubinActions structure RubinAction (G α : Type _) extends Group G, TopologicalSpace α, MulAction G α, FaithfulSMul G α where locally_compact : LocallyCompactSpace α hausdorff : T2Space α no_isolated_points : HasNoIsolatedPoints α locallyDense : LocallyDense G α #align rubin_action Rubin.RubinAction end RubinActions section AlgebraicDisjointness variable {G α : Type _} variable [TopologicalSpace α] variable [Group G] variable [MulAction G α] 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_mono 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 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_mono 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_mono 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_mono 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 RegularSupport lemma lemma_2_2 (G: Type _) {α : Type _} [Group G] [TopologicalSpace α] [MulAction G 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) : Monoid.exponent G•[U] = 0 := by by_contra exp_ne_zero let ⟨p, ⟨g, g_in_ristU⟩, n, p_in_U, n_pos, hpgn, n_eq_Sup⟩ := Period.period_from_exponent U U_nonempty exp_ne_zero simp at hpgn let ⟨V', V'_open, p_in_V', disj'⟩ := disjoint_nbhd_fin (smul_injective_within_period hpgn) let V := U ∩ V' have V_open : IsOpen V := U_open.inter V'_open have p_in_V : p ∈ V := ⟨p_in_U, p_in_V'⟩ have disj : ∀ (i j : Fin n), i ≠ j → Disjoint (g ^ (i : ℕ) •'' V) (g ^ (j : ℕ) •'' V) := by intro i j i_ne_j apply Set.disjoint_of_subset · apply smulImage_mono apply Set.inter_subset_right · apply smulImage_mono apply Set.inter_subset_right exact disj' i j i_ne_j let ⟨h, h_in_ristV, h_ne_one⟩ := h_lm.get_nontrivial_rist_elem V_open (Set.nonempty_of_mem p_in_V) have hg_in_ristU : h * g ∈ RigidStabilizer G U := by simp [RigidStabilizer] intro x x_notin_U rw [mul_smul] rw [g_in_ristU _ x_notin_U] have x_notin_V : x ∉ V := fun x_in_V => x_notin_U x_in_V.left rw [h_in_ristV _ x_notin_V] let ⟨q, q_in_V, hq_ne_q ⟩ := faithful_rigid_stabilizer_moves_point h_in_ristV h_ne_one have gpowi_q_notin_V : ∀ (i : Fin n), (i : ℕ) > 0 → g ^ (i : ℕ) • q ∉ V := by apply smulImage_distinct_of_disjoint_pow n_pos disj exact q_in_V -- We have (hg)^i q = g^i q for all 0 < i < n have hgpow_eq_gpow : ∀ (i : Fin n), (h * g) ^ (i : ℕ) • q = g ^ (i : ℕ) • q := by intro ⟨i, i_lt_n⟩ simp induction i with | zero => simp | succ i' IH => have i'_lt_n: i' < n := Nat.lt_of_succ_lt i_lt_n have IH := IH i'_lt_n rw [smul_succ] rw [IH] rw [smul_succ] rw [mul_smul] rw [<-smul_succ] -- We can show that `g^(Nat.succ i') • q ∉ V`, -- which means that with `h` in `RigidStabilizer G V`, `h` fixes that point apply h_in_ristV (g^(Nat.succ i') • q) let i'₂ : Fin n := ⟨Nat.succ i', i_lt_n⟩ have h_eq: Nat.succ i' = (i'₂ : ℕ) := by simp rw [h_eq] apply smulImage_distinct_of_disjoint_pow · exact n_pos · exact disj · exact q_in_V · simp -- Combined with `g^i • q ≠ q`, this yields `(hg)^i • q ≠ q` for all `0 < i < n` have hgpow_moves : ∀ (i : Fin n), 0 < (i : ℕ) → (h*g)^(i : ℕ) • q ≠ q := by intro ⟨i, i_lt_n⟩ i_pos simp at i_pos rw [hgpow_eq_gpow] simp by_contra h' apply gpowi_q_notin_V ⟨i, i_lt_n⟩ exact i_pos simp (config := {zeta := false}) only [] rw [h'] exact q_in_V -- This even holds for `i = n` have hgpown_moves : (h * g) ^ n • q ≠ q := by -- Rewrite (hg)^n • q = h * g^n • q rw [<-Nat.succ_pred n_pos.ne.symm] rw [pow_succ] have h_eq := hgpow_eq_gpow ⟨Nat.pred n, Nat.pred_lt_self n_pos⟩ simp at h_eq rw [mul_smul, h_eq, <-mul_smul, mul_assoc, <-pow_succ] rw [<-Nat.succ_eq_add_one, Nat.succ_pred n_pos.ne.symm] -- We first eliminate `g^n • q` by proving that `n = Period g q` have period_gq_eq_n : Period.period q g = n := by apply ge_antisymm { apply Period.notfix_le_period' · exact n_pos · apply Period.period_pos' · exact Set.nonempty_of_mem p_in_U · exact exp_ne_zero · exact q_in_V.left · exact g_in_ristU · intro i i_pos rw [<-hgpow_eq_gpow] apply hgpow_moves i i_pos } { rw [n_eq_Sup] apply Period.period_le_Sup_periods' · exact Set.nonempty_of_mem p_in_U · exact exp_ne_zero · exact q_in_V.left · exact g_in_ristU } rw [mul_smul, <-period_gq_eq_n] rw [Period.pow_period_fix] -- Finally, we have `h • q ≠ q` exact hq_ne_q -- Finally, we derive a contradiction have ⟨period_hg_pos, period_hg_le_n⟩ := Period.zero_lt_period_le_Sup_periods U_nonempty exp_ne_zero ⟨q, q_in_V.left⟩ ⟨h * g, hg_in_ristU⟩ simp at period_hg_pos simp at period_hg_le_n rw [<-n_eq_Sup] at period_hg_le_n cases (lt_or_eq_of_le period_hg_le_n) with | inl period_hg_lt_n => apply hgpow_moves ⟨Period.period q (h * g), period_hg_lt_n⟩ exact period_hg_pos simp apply Period.pow_period_fix | inr period_hg_eq_n => apply hgpown_moves rw [<-period_hg_eq_n] apply Period.pow_period_fix -- Given the statement `¬Support α h ⊆ RegularSupport α f`, -- we construct an open subset within `Support α h \ RegularSupport α f`, -- and we show that it is non-empty, open and (by construction) disjoint from `Support α f`. lemma open_set_from_supp_not_subset_rsupp {G α : Type _} [Group G] [TopologicalSpace α] [MulAction G α] [ContinuousMulAction G α] [T2Space α] {f h : G} (not_support_subset_rsupp : ¬Support α h ⊆ RegularSupport α f): ∃ V : Set α, V ⊆ Support α h ∧ Set.Nonempty V ∧ IsOpen V ∧ Disjoint V (Support α f) := by let U := Support α h \ closure (RegularSupport α f) have U_open : IsOpen U := by unfold_let rw [Set.diff_eq_compl_inter] apply IsOpen.inter · simp · exact support_open _ have U_subset_supp_h : U ⊆ Support α h := by simp; apply Set.diff_subset have U_disj_supp_f : Disjoint U (Support α f) := by apply Set.disjoint_of_subset_right · exact subset_closure · simp rw [Set.diff_eq_compl_inter] apply Disjoint.inter_left apply Disjoint.closure_right; swap; simp rw [Set.disjoint_compl_left_iff_subset] apply subset_trans exact subset_closure apply closure_mono apply support_subset_regularSupport have U_nonempty : Set.Nonempty U; swap exact ⟨U, U_subset_supp_h, U_nonempty, U_open, U_disj_supp_f⟩ -- We prove that U isn't empty by contradiction: -- if it is empty, then `Support α h \ closure (RegularSupport α f) = ∅`, -- so we can show that `Support α h ⊆ RegularSupport α f`, -- contradicting with our initial hypothesis. by_contra U_empty apply not_support_subset_rsupp show Support α h ⊆ RegularSupport α f apply subset_from_diff_closure_eq_empty · apply regularSupport_regular · exact support_open _ · rw [Set.not_nonempty_iff_eq_empty] at U_empty exact U_empty lemma nontrivial_pow_from_exponent_eq_zero {G : Type _} [Group G] (exp_eq_zero : Monoid.exponent G = 0) : ∀ (n : ℕ), n > 0 → ∃ g : G, g^n ≠ 1 := by intro n n_pos rw [Monoid.exponent_eq_zero_iff] at exp_eq_zero unfold Monoid.ExponentExists at exp_eq_zero rw [<-Classical.not_forall_not, Classical.not_not] at exp_eq_zero simp at exp_eq_zero exact exp_eq_zero n n_pos lemma proposition_2_1 {G α : Type _} [Group G] [TopologicalSpace α] [MulAction G α] [ContinuousMulAction G α] [T2Space α] [LocallyMoving G α] [h_faithful : FaithfulSMul G α] (f : G) : AlgebraicCentralizer f = G•[RegularSupport α f] := by ext h constructor swap { intro h_in_rist simp at h_in_rist unfold AlgebraicCentralizer rw [Subgroup.mem_centralizer_iff] intro g g_in_S simp [AlgebraicSubgroup] at g_in_S let ⟨g', ⟨g'_alg_disj, g_eq_g'⟩⟩ := g_in_S have supp_disj := proposition_1_1_2 f g' g'_alg_disj (α := α) apply Commute.eq symm apply commute_if_rigidStabilizer_and_disjoint (α := α) · exact h_in_rist · show Disjoint (RegularSupport α f) (Support α g) have cl_supp_disj : Disjoint (closure (Support α f)) (Support α g) swap apply Set.disjoint_of_subset _ _ cl_supp_disj · rw [RegularSupport.def] exact interior_subset · rfl · rw [<-g_eq_g'] exact Disjoint.closure_left supp_disj (support_open _) } intro h_in_centralizer by_contra h_notin_rist simp at h_notin_rist rw [rigidStabilizer_support] at h_notin_rist let ⟨V, V_in_supp_h, V_nonempty, V_open, V_disj_supp_f⟩ := open_set_from_supp_not_subset_rsupp h_notin_rist let ⟨v, v_in_V⟩ := V_nonempty have v_moved := V_in_supp_h v_in_V rw [mem_support] at v_moved have ⟨W, W_open, v_in_W, W_subset_support, disj_W_img⟩ := disjoint_nbhd_in V_open v_in_V v_moved have mono_exp := lemma_2_2 G W_open (Set.nonempty_of_mem v_in_W) let ⟨⟨g, g_in_rist⟩, g12_ne_one⟩ := nontrivial_pow_from_exponent_eq_zero mono_exp 12 (by norm_num) simp at g12_ne_one have disj_supports : Disjoint (Support α f) (Support α g) := by apply Set.disjoint_of_subset_right · apply rigidStabilizer_support.mp exact g_in_rist · apply Set.disjoint_of_subset_right · exact W_subset_support · exact V_disj_supp_f.symm have alg_disj_f_g := proposition_1_1_1 _ _ disj_supports have g12_in_algebraic_subgroup : g^12 ∈ AlgebraicSubgroup f := by simp [AlgebraicSubgroup] use g constructor exact ↑alg_disj_f_g rfl have h_nc_g12 : ¬Commute (g^12) h := by have supp_g12_sub_W : Support α (g^12) ⊆ W := by rw [rigidStabilizer_support] at g_in_rist calc Support α (g^12) ⊆ Support α g := by apply support_pow _ ⊆ W := g_in_rist have supp_g12_disj_hW : Disjoint (Support α (g^12)) (h •'' W) := by apply Set.disjoint_of_subset_left swap · exact disj_W_img · exact supp_g12_sub_W exact not_commute_of_disj_support_smulImage g12_ne_one supp_g12_sub_W supp_g12_disj_hW apply h_nc_g12 exact h_in_centralizer _ g12_in_algebraic_subgroup -- Small lemma for remark 2.3 theorem rigidStabilizer_inter_bot_iff_regularSupport_disj {G α : Type _} [Group G] [TopologicalSpace α] [MulAction G α] [ContinuousMulAction G α] [LocallyMoving G α] [FaithfulSMul G α] {f g : G} : G•[RegularSupport α f] ⊓ G•[RegularSupport α g] = ⊥ ↔ Disjoint (RegularSupport α f) (RegularSupport α g) := by rw [<-rigidStabilizer_inter] constructor { intro rist_disj by_contra rsupp_not_disj rw [Set.not_disjoint_iff] at rsupp_not_disj let ⟨x, x_in_rsupp_f, x_in_rsupp_g⟩ := rsupp_not_disj have rsupp_open: IsOpen (RegularSupport α f ∩ RegularSupport α g) := by apply IsOpen.inter <;> exact regularSupport_open _ _ -- The contradiction occurs by applying the definition of LocallyMoving: apply LocallyMoving.locally_moving (G := G) _ rsupp_open _ rist_disj exact ⟨x, x_in_rsupp_f, x_in_rsupp_g⟩ } { intro rsupp_disj rw [Set.disjoint_iff_inter_eq_empty] at rsupp_disj rw [rsupp_disj] by_contra rist_ne_bot rw [<-ne_eq, Subgroup.ne_bot_iff_exists_ne_one] at rist_ne_bot let ⟨⟨h, h_in_rist⟩, h_ne_one⟩ := rist_ne_bot simp at h_ne_one apply h_ne_one rw [rigidStabilizer_empty] at h_in_rist rw [Subgroup.mem_bot] at h_in_rist exact h_in_rist } variable {G α : Type _} variable [Group G] variable [TopologicalSpace α] [T2Space α] variable [MulAction G α] [ContinuousMulAction G α] [FaithfulSMul G α] [LocallyMoving G α] /-- This demonstrates that the disjointness of the supports of two elements `f` and `g` can be proven without knowing anything about how `f` and `g` act on `α` (bar the more global properties of the group action). We could prove that the intersection of the algebraic centralizers of `f` and `g` is trivial purely within group theory, and then apply this theorem to know that their support in `α` will be disjoint. --/ lemma remark_2_3 {f g : G} : (AlgebraicCentralizer f) ⊓ (AlgebraicCentralizer g) = ⊥ → Disjoint (Support α f) (Support α g) := by intro alg_disj rw [disjoint_interiorClosure_iff (support_open _) (support_open _)] simp repeat rw [<-RegularSupport.def] rw [<-rigidStabilizer_inter_bot_iff_regularSupport_disj] repeat rw [<-proposition_2_1] exact alg_disj #check proposition_2_1 lemma rigidStabilizerInter_eq_algebraicCentralizerInter {S : Finset G} : RigidStabilizerInter₀ α S = AlgebraicCentralizerInter₀ S := by unfold RigidStabilizerInter₀ unfold AlgebraicCentralizerInter₀ simp only [<-proposition_2_1] -- conv => { -- rhs -- congr; intro; congr; intro -- rw [proposition_2_1 (α := α)] -- } theorem rigidStabilizerBasis_eq_algebraicCentralizerBasis : AlgebraicCentralizerBasis G = RigidStabilizerBasis G α := by apply le_antisymm <;> intro B B_mem any_goals rw [RigidStabilizerBasis.mem_iff] any_goals rw [AlgebraicCentralizerBasis.mem_iff] any_goals rw [RigidStabilizerBasis.mem_iff] at B_mem any_goals rw [AlgebraicCentralizerBasis.mem_iff] at B_mem all_goals let ⟨⟨seed, B_ne_bot⟩, B_eq⟩ := B_mem any_goals rw [RigidStabilizerBasis₀.val_def] at B_eq any_goals rw [AlgebraicCentralizerBasis₀.val_def] at B_eq all_goals simp at B_eq all_goals rw [<-B_eq] rw [<-rigidStabilizerInter_eq_algebraicCentralizerInter (α := α)] at B_ne_bot swap rw [rigidStabilizerInter_eq_algebraicCentralizerInter (α := α)] at B_ne_bot all_goals use ⟨seed, B_ne_bot⟩ symm all_goals apply rigidStabilizerInter_eq_algebraicCentralizerInter end RegularSupport section HomeoGroup open Topology variable {G α : Type _} [Group G] [TopologicalSpace α] [T2Space α] variable [MulAction G α] [ContinuousMulAction G α] [FaithfulSMul G α] [LocallyMoving G α] example : TopologicalSpace G := TopologicalSpace.generateFrom (RigidStabilizerBasis.asSets G α) theorem proposition_3_4_2 {α : Type _} [TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] (F : Ultrafilter α): (∃ p : α, ClusterPt p F) ↔ ∃ S ∈ F, IsCompact (closure S) := by constructor · intro ⟨p, p_clusterPt⟩ rw [Ultrafilter.clusterPt_iff] at p_clusterPt have ⟨S, S_in_nhds, S_compact⟩ := (compact_basis_nhds p).ex_mem use S constructor exact p_clusterPt S_in_nhds rw [IsClosed.closure_eq S_compact.isClosed] exact S_compact · intro ⟨S, S_in_F, clS_compact⟩ have F_le_principal_S : F ≤ Filter.principal (closure S) := by rw [Filter.le_principal_iff] simp apply Filter.sets_of_superset exact S_in_F exact subset_closure let ⟨x, _, F_clusterPt⟩ := clS_compact F_le_principal_S use x end HomeoGroup section Ultrafilter variable {G α : Type _} variable [Group G] variable [TopologicalSpace α] [T2Space α] variable [MulAction G α] [ContinuousMulAction G α] [FaithfulSMul G α] [LocallyMoving G α] def RSuppSubsets {α : Type _} [TopologicalSpace α] (V : Set α) : Set (Set α) := {W ∈ RegularSupportBasis.asSet α | W ⊆ V} def RSuppOrbit {G α : Type _} [Group G] [TopologicalSpace α] [MulAction G α] (F : Filter α) (H : Subgroup G) : Set (Set α) := { g •'' W | (g ∈ H) (W ∈ F) } lemma moving_elem_of_open_subset_closure_orbit {U V : Set α} (U_open : IsOpen U) (U_nonempty : Set.Nonempty U) {p : α} (U_ss_clOrbit : U ⊆ closure (MulAction.orbit G•[V] p)) : ∃ h : G, h ∈ G•[V] ∧ h • p ∈ U := by have p_in_orbit : p ∈ MulAction.orbit G•[V] p := by simp have ⟨q, ⟨q_in_U, q_in_orbit⟩⟩ := inter_of_open_subset_of_closure U_open U_nonempty ⟨p, p_in_orbit⟩ U_ss_clOrbit rw [MulAction.mem_orbit_iff] at q_in_orbit let ⟨⟨h, h_in_orbit⟩, hq_eq_p⟩ := q_in_orbit simp at hq_eq_p use h constructor assumption rw [hq_eq_p] assumption lemma compact_subset_of_rsupp_basis (G : Type _) {α : Type _} [Group G] [TopologicalSpace α] [T2Space α] [MulAction G α] [ContinuousMulAction G α] [LocallyCompactSpace α] [HasNoIsolatedPoints α] [LocallyDense G α] (U : RegularSupportBasis α): ∃ V : RegularSupportBasis α, (closure V.val) ⊆ U.val ∧ IsCompact (closure V.val) := by let ⟨x, x_in_U⟩ := U.nonempty let ⟨W, W_compact, x_in_intW, W_ss_U⟩ := exists_compact_subset U.regular.isOpen x_in_U have ⟨V, V_in_basis, x_in_V, V_ss_intW⟩ := (RegularSupportBasis.isBasis G α).exists_subset_of_mem_open x_in_intW isOpen_interior have clV_ss_W : closure V ⊆ W := by calc closure V ⊆ closure (interior W) := by apply closure_mono exact V_ss_intW _ ⊆ closure W := by apply closure_mono exact interior_subset _ = W := by apply IsClosed.closure_eq exact W_compact.isClosed rw [RegularSupportBasis.mem_asSet] at V_in_basis let ⟨V', V'_val⟩ := V_in_basis use V' rw [V'_val] constructor · exact subset_trans clV_ss_W W_ss_U · exact IsCompact.of_isClosed_subset W_compact isClosed_closure clV_ss_W /-- # Proposition 3.5 This proposition gives an alternative definition for an ultrafilter to converge within a set `U`. This alternative definition should be reconstructible entirely from the algebraic structure of `G`. --/ theorem proposition_3_5 [LocallyDense G α] [LocallyCompactSpace α] [HasNoIsolatedPoints α] (U : RegularSupportBasis α) (F: Ultrafilter α): (∃ p ∈ U.val, ClusterPt p F) ↔ ∃ V : RegularSupportBasis α, V ≤ U ∧ RSuppSubsets V.val ⊆ RSuppOrbit F G•[U.val] := by constructor { simp intro p p_in_U p_clusterPt have U_open : IsOpen U.val := U.regular.isOpen -- First, get a neighborhood of p that is a subset of the closure of the orbit of G_U have clOrbit_in_nhds := LocallyDense.rigidStabilizer_in_nhds G α U_open p_in_U rw [mem_nhds_iff] at clOrbit_in_nhds let ⟨V, V_ss_clOrbit, V_open, p_in_V⟩ := clOrbit_in_nhds clear clOrbit_in_nhds -- Then, get a nontrivial element from that set let ⟨g, g_in_rist, g_ne_one⟩ := LocallyMoving.get_nontrivial_rist_elem (G := G) V_open ⟨p, p_in_V⟩ have V_ss_clU : V ⊆ closure U.val := by apply subset_trans exact V_ss_clOrbit apply closure_mono exact orbit_rigidStabilizer_subset p_in_U -- The regular support of g is within U have rsupp_ss_U : RegularSupport α g ⊆ U.val := by rw [RegularSupport] rw [rigidStabilizer_support] at g_in_rist calc InteriorClosure (Support α g) ⊆ InteriorClosure V := by apply interiorClosure_mono assumption _ ⊆ InteriorClosure (closure U.val) := by apply interiorClosure_mono assumption _ ⊆ InteriorClosure U.val := by simp rfl _ ⊆ _ := by apply subset_of_eq exact U.regular -- Use as the chosen set RegularSupport g let g' : HomeoGroup α := HomeoGroup.fromContinuous α g have g'_ne_one : g' ≠ 1 := by simp rw [<-HomeoGroup.fromContinuous_one (G := G)] rw [HomeoGroup.fromContinuous_eq_iff] exact g_ne_one use RegularSupportBasis.fromSingleton g' g'_ne_one constructor · -- This statement is equivalent to rsupp(g) ⊆ U rw [RegularSupportBasis.le_def] rw [RegularSupportBasis.fromSingleton_val] unfold RegularSupportInter₀ simp exact rsupp_ss_U · intro W W_in_subsets rw [RegularSupportBasis.fromSingleton_val] at W_in_subsets unfold RSuppSubsets RegularSupportInter₀ at W_in_subsets simp at W_in_subsets let ⟨W_in_basis, W_subset_rsupp⟩ := W_in_subsets clear W_in_subsets g' g'_ne_one unfold RSuppOrbit simp -- We have that W is a subset of the closure of the orbit of G_U have W_ss_clOrbit : W ⊆ closure (MulAction.orbit (↥(RigidStabilizer G U.val)) p) := by rw [rigidStabilizer_support] at g_in_rist calc W ⊆ RegularSupport α g := by assumption _ ⊆ closure (Support α g) := regularSupport_subset_closure_support _ ⊆ closure V := by apply closure_mono assumption _ ⊆ _ := by rw [<-closure_closure (s := MulAction.orbit _ _)] apply closure_mono assumption -- W is also open and nonempty... have W_open : IsOpen W := by let ⟨W', W'_eq⟩ := (RegularSupportBasis.mem_asSet _).mp W_in_basis rw [<-W'_eq] exact W'.regular.isOpen have W_nonempty : Set.Nonempty W := by let ⟨W', W'_eq⟩ := (RegularSupportBasis.mem_asSet _).mp W_in_basis rw [<-W'_eq] exact W'.nonempty -- So we can get an element `h` such that `h • p ∈ W` and `h ∈ G_U` let ⟨h, h_in_rist, hp_in_W⟩ := moving_elem_of_open_subset_closure_orbit W_open W_nonempty W_ss_clOrbit use h constructor exact h_in_rist use h⁻¹ •'' W constructor swap { rw [smulImage_mul] simp } -- We just need to show that h⁻¹ •'' W ∈ F, that is, h⁻¹ •'' W ∈ 𝓝 p rw [Ultrafilter.clusterPt_iff] at p_clusterPt apply p_clusterPt have basis := (RegularSupportBasis.isBasis G α).nhds_hasBasis (a := p) rw [basis.mem_iff] use h⁻¹ •'' W repeat' apply And.intro · rw [RegularSupportBasis.mem_asSet] rw [RegularSupportBasis.mem_asSet] at W_in_basis let ⟨W', W'_eq⟩ := W_in_basis have dec_eq : DecidableEq (HomeoGroup α) := Classical.decEq _ use (HomeoGroup.fromContinuous α h⁻¹) • W' rw [RegularSupportBasis.smul_val, W'_eq] simp · rw [mem_smulImage, inv_inv] exact hp_in_W · exact Eq.subset rfl } { intro ⟨V, ⟨V_ss_U, subsets_ss_orbit⟩⟩ rw [RegularSupportBasis.le_def] at V_ss_U -- Obtain a compact subset of V' in the basis let ⟨V', clV'_ss_V, clV'_compact⟩ := compact_subset_of_rsupp_basis G V have V'_in_subsets : V'.val ∈ RSuppSubsets V.val := by unfold RSuppSubsets simp constructor unfold RegularSupportBasis.asSet simp exact subset_trans subset_closure clV'_ss_V -- V' is in the orbit, so there exists a value `g ∈ G_U` such that `gV ∈ F` -- Note that with the way we set up the equations, we obtain `g⁻¹` have V'_in_orbit := subsets_ss_orbit V'_in_subsets simp [RSuppOrbit] at V'_in_orbit let ⟨g, g_in_rist, ⟨W, W_in_F, gW_eq_V⟩⟩ := V'_in_orbit have gV'_in_F : g⁻¹ •'' V' ∈ F := by rw [smulImage_inv] at gW_eq_V rw [<-gW_eq_V] assumption have gV'_compact : IsCompact (closure (g⁻¹ •'' V'.val)) := by rw [smulImage_closure] apply smulImage_compact assumption have ⟨p, p_lim⟩ := (proposition_3_4_2 F).mpr ⟨g⁻¹ •'' V'.val, ⟨gV'_in_F, gV'_compact⟩⟩ use p constructor swap assumption rw [clusterPt_iff_forall_mem_closure] at p_lim specialize p_lim (g⁻¹ •'' V') gV'_in_F rw [smulImage_closure, mem_smulImage, inv_inv] at p_lim rw [rigidStabilizer_support, <-support_inv] at g_in_rist rw [<-fixed_smulImage_in_support g⁻¹ g_in_rist] rw [mem_smulImage, inv_inv] apply V_ss_U apply clV'_ss_V exact p_lim } end Ultrafilter variable {G α : Type _} variable [Group G] variable [TopologicalSpace α] [MulAction G α] [ContinuousMulAction G α] def IsRigidSubgroup (S : Set G) := S ≠ {1} ∧ ∃ T : Finset G, S = ⨅ (f ∈ T), AlgebraicCentralizer f def IsRigidSubgroup.toSubgroup {S : Set G} (S_rigid : IsRigidSubgroup S) : Subgroup G where carrier := S mul_mem' := by let ⟨_, T, S_eq⟩ := S_rigid simp only [S_eq, SetLike.mem_coe] apply Subgroup.mul_mem one_mem' := by let ⟨_, T, S_eq⟩ := S_rigid simp only [S_eq, SetLike.mem_coe] apply Subgroup.one_mem inv_mem' := by let ⟨_, T, S_eq⟩ := S_rigid simp only [S_eq, SetLike.mem_coe] apply Subgroup.inv_mem theorem IsRigidSubgroup.mem_subgroup {S : Set G} (S_rigid : IsRigidSubgroup S) (g : G): g ∈ S ↔ g ∈ S_rigid.toSubgroup := by rfl theorem IsRigidSubgroup.toSubgroup_neBot {S : Set G} (S_rigid : IsRigidSubgroup S) : S_rigid.toSubgroup ≠ ⊥ := by intro eq_bot rw [Subgroup.eq_bot_iff_forall] at eq_bot simp only [<-mem_subgroup] at eq_bot apply S_rigid.left rw [Set.eq_singleton_iff_unique_mem] constructor · let ⟨S', S'_eq⟩ := S_rigid.right rw [S'_eq, SetLike.mem_coe] exact Subgroup.one_mem _ · assumption lemma Subgroup.coe_eq (S T : Subgroup G) : (S : Set G) = (T : Set G) ↔ S = T := by constructor · intro h ext x repeat rw [<-Subgroup.mem_carrier] have h₁ : ∀ S : Subgroup G, (S : Set G) = S.carrier := by intro h; rfl repeat rw [h₁] at h rw [h] · intro h rw [h] def IsRigidSubgroup.algebraicCentralizerBasis {S : Set G} (S_rigid : IsRigidSubgroup S) : AlgebraicCentralizerBasis G := ⟨ S_rigid.toSubgroup, by rw [AlgebraicCentralizerBasis.mem_iff' _ (IsRigidSubgroup.toSubgroup_neBot S_rigid)] let ⟨S', S'_eq⟩ := S_rigid.right use S' unfold AlgebraicCentralizerInter₀ rw [<-Subgroup.coe_eq, <-S'_eq] rfl ⟩ theorem IsRigidSubgroup.algebraicCentralizerBasis_val {S : Set G} (S_rigid : IsRigidSubgroup S) : S_rigid.algebraicCentralizerBasis.val = S_rigid.toSubgroup := rfl section toRegularSupportBasis variable (α : Type _) variable [TopologicalSpace α] [MulAction G α] [ContinuousMulAction G α] variable [FaithfulSMul G α] [T2Space α] [LocallyMoving G α] theorem IsRigidSubgroup.has_regularSupportBasis {S : Set G} (S_rigid : IsRigidSubgroup S) : ∃ (U : RegularSupportBasis α), G•[U.val] = S := by let ⟨S_ne_bot, ⟨T, S_eq⟩⟩ := S_rigid rw [S_eq] simp only [Subgroup.coe_eq] rw [S_eq, <-Subgroup.coe_bot, ne_eq, Subgroup.coe_eq, <-ne_eq] at S_ne_bot let T' : Finset (HomeoGroup α) := Finset.map (HomeoGroup.fromContinuous_embedding α) T have T'_rsupp_nonempty : Set.Nonempty (RegularSupportInter₀ T') := by unfold RegularSupportInter₀ simp only [proposition_2_1 (G := G) (α := α)] at S_ne_bot rw [ne_eq, <-rigidStabilizer_iInter_regularSupport', <-ne_eq] at S_ne_bot let ⟨x, x_in_iInter⟩ := rigidStabilizer_neBot S_ne_bot use x simp simp at x_in_iInter exact x_in_iInter let T'' := RegularSupportBasis.fromSeed ⟨T', T'_rsupp_nonempty⟩ have T''_val : T''.val = RegularSupportInter₀ T' := rfl use T'' rw [T''_val] unfold RegularSupportInter₀ simp rw [rigidStabilizer_iInter_regularSupport'] simp only [<-proposition_2_1] noncomputable def IsRigidSubgroup.toRegularSupportBasis {S : Set G} (S_rigid : IsRigidSubgroup S) : RegularSupportBasis α := Exists.choose (IsRigidSubgroup.has_regularSupportBasis α S_rigid) theorem IsRigidSubgroup.toRegularSupportBasis_eq {S : Set G} (S_rigid : IsRigidSubgroup S): G•[(S_rigid.toRegularSupportBasis α).val] = S := by exact Exists.choose_spec (IsRigidSubgroup.has_regularSupportBasis α S_rigid) theorem IsRigidSubgroup.toRegularSupportBasis_mono {S T : Set G} (S_rigid : IsRigidSubgroup S) (T_rigid : IsRigidSubgroup T) : S ⊆ T ↔ S_rigid.toRegularSupportBasis α ≤ T_rigid.toRegularSupportBasis α := by rw [RegularSupportBasis.le_def] constructor · intro S_ss_T rw [<-IsRigidSubgroup.toRegularSupportBasis_eq (α := α) S_rigid] at S_ss_T rw [<-IsRigidSubgroup.toRegularSupportBasis_eq (α := α) T_rigid] at S_ss_T simp at S_ss_T rw [<-rigidStabilizer_subset_iff G (RegularSupportBasis.regular _) (RegularSupportBasis.regular _)] at S_ss_T exact S_ss_T · intro Sr_ss_Tr rw [rigidStabilizer_subset_iff G (RegularSupportBasis.regular _) (RegularSupportBasis.regular _)] at Sr_ss_Tr -- TODO: clean that up have Sr_ss_Tr' : (G•[(toRegularSupportBasis α S_rigid).val] : Set G) ⊆ G•[(toRegularSupportBasis α T_rigid).val] := by simp assumption rw [IsRigidSubgroup.toRegularSupportBasis_eq (α := α) S_rigid] at Sr_ss_Tr' rw [IsRigidSubgroup.toRegularSupportBasis_eq (α := α) T_rigid] at Sr_ss_Tr' assumption theorem IsRigidSubgroup.toRegularSupportBasis_mono' {S T : Set G} (S_rigid : IsRigidSubgroup S) (T_rigid : IsRigidSubgroup T) : S ⊆ T ↔ (S_rigid.toRegularSupportBasis α : Set α) ⊆ (T_rigid.toRegularSupportBasis α : Set α) := by rw [<-RegularSupportBasis.le_def] rw [<-IsRigidSubgroup.toRegularSupportBasis_mono] end toRegularSupportBasis theorem IsRigidSubgroup.conj {U : Set G} (U_rigid : IsRigidSubgroup U) (g : G) : IsRigidSubgroup ((fun h => g * h * g⁻¹) '' U) := by have conj_bijective : ∀ g : G, Function.Bijective (fun h => g * h * g⁻¹) := by intro g constructor · intro x y; simp · intro x use g⁻¹ * x * g group constructor · intro H apply U_rigid.left have h₁ : (fun h => g * h * g⁻¹) '' {1} = {1} := by simp rw [<-h₁] at H apply (Set.image_eq_image (conj_bijective g).left).mp H · let ⟨S, S_eq⟩ := U_rigid.right have dec_eq : DecidableEq G := Classical.typeDecidableEq G use Finset.image (fun h => g * h * g⁻¹) S rw [S_eq] simp simp only [Set.image_iInter (conj_bijective _), AlgebraicCentralizer.conj] def AlgebraicSubsets (V : Set G) : Set (Subgroup G) := {W ∈ AlgebraicCentralizerBasis G | W ≤ V} def AlgebraicOrbit (F : Ultrafilter G) (U : Set G) : Set (Subgroup G) := { (W_rigid.conj g).toSubgroup | (g ∈ U) (W ∈ F) (W_rigid : IsRigidSubgroup W) } structure RubinFilter (G : Type _) [Group G] where filter : Ultrafilter G -- rigid_basis : ∀ S ∈ filter, ∃ T ⊆ S, IsRigidSubgroup T converges : ∀ U ∈ filter, IsRigidSubgroup U → ∃ V : Set G, IsRigidSubgroup V ∧ V ⊆ U ∧ AlgebraicSubsets V ⊆ AlgebraicOrbit filter U -- Only really used to prove that ∀ S : Rigid, T : Rigid, S T ∈ F, S ∩ T : Rigid ne_bot : {1} ∉ filter instance : Coe (RubinFilter G) (Ultrafilter G) where coe := RubinFilter.filter section Equivalence open Topology variable (α : Type _) variable [TopologicalSpace α] [MulAction G α] [ContinuousMulAction G α] variable [FaithfulSMul G α] [T2Space α] [LocallyMoving G α] -- TODO: either see whether we actually need this step, or change these names to something memorable -- This is an attempt to convert a RubinFilter G back to an Ultrafilter α def RubinFilter.to_action_filter (F : RubinFilter G) : Filter α := ⨅ (S : { S : Set G // S ∈ F.filter ∧ IsRigidSubgroup S }), (Filter.principal (S.prop.right.toRegularSupportBasis α)) instance RubinFilter.to_action_filter_neBot {F : RubinFilter G} [Nonempty α] : Filter.NeBot (F.to_action_filter α) := by unfold to_action_filter rw [Filter.iInf_neBot_iff_of_directed] · intro ⟨S, S_in_F, S_rigid⟩ simp · intro ⟨S, S_in_F, S_rigid⟩ ⟨T, T_in_F, T_rigid⟩ simp use S ∩ T have ST_in_F : (S ∩ T) ∈ F.filter := by rw [<-Ultrafilter.mem_coe] apply Filter.inter_mem <;> assumption have ST_subgroup : IsRigidSubgroup (S ∩ T) := by constructor swap · let ⟨S_seed, S_eq⟩ := S_rigid.right let ⟨T_seed, T_eq⟩ := T_rigid.right have dec_eq : DecidableEq G := Classical.typeDecidableEq G use S_seed ∪ T_seed rw [Finset.iInf_union, S_eq, T_eq] simp · -- TODO: check if we can't prove this without using F.ne_bot; -- we might be able to use convergence intro ST_eq_bot apply F.ne_bot rw [<-ST_eq_bot] exact ST_in_F -- sorry use ⟨ST_in_F, ST_subgroup⟩ repeat rw [<-IsRigidSubgroup.toRegularSupportBasis_mono' (α := α)] constructor exact Set.inter_subset_left S T exact Set.inter_subset_right S T -- lemma RubinFilter.mem_to_action_filter' (F : RubinFilter G) (U : Set α) : -- U ∈ F.to_action_filter α ↔ ∃ S : AlgebraicCentralizerBasis G, S.val ∈ F.filter, lemma RubinFilter.mem_to_action_filter (F : RubinFilter G) {U : Set G} (U_rigid : IsRigidSubgroup U) : U ∈ F.filter ↔ (U_rigid.toRegularSupportBasis α : Set α) ∈ F.to_action_filter α := by unfold to_action_filter constructor · intro U_in_filter apply Filter.mem_iInf_of_mem ⟨U, U_in_filter, U_rigid⟩ intro x simp · sorry noncomputable def RubinFilter.to_action_ultrafilter (F : RubinFilter G) [Nonempty α]: Ultrafilter α := Ultrafilter.of (F.to_action_filter α) theorem RubinFilter.to_action_ultrafilter_converges (F : RubinFilter G) [Nonempty α] [LocallyDense G α] [HasNoIsolatedPoints α] [LocallyCompactSpace α] {U : Set G} (U_in_F : U ∈ F.filter) (U_rigid : IsRigidSubgroup U): ∃ p ∈ (U_rigid.toRegularSupportBasis α).val, ClusterPt p (F.to_action_ultrafilter α) := by rw [proposition_3_5 (G := G)] let ⟨V, V_rigid, V_ss_U, algsubs_ss_algorb⟩ := F.converges U U_in_F U_rigid let V' := V_rigid.toSubgroup -- TODO: subst V' to simplify the proof? use V_rigid.toRegularSupportBasis α constructor { rw [<-IsRigidSubgroup.toRegularSupportBasis_mono] exact V_ss_U } unfold RSuppSubsets RSuppOrbit simp intro S S_rsupp_basis S_ss_V have S_regular : Regular S := by let ⟨S', S'_eq⟩ := (RegularSupportBasis.mem_asSet S).mp S_rsupp_basis rw [<-S'_eq] exact S'.regular have S_nonempty : Set.Nonempty S := by let ⟨S', S'_eq⟩ := (RegularSupportBasis.mem_asSet S).mp S_rsupp_basis rw [<-S'_eq] exact S'.nonempty have GS_ss_V : G•[S] ≤ V := by rw [<-V_rigid.toRegularSupportBasis_eq (α := α)] simp rw [<-rigidStabilizer_subset_iff G (α := α) S_regular (RegularSupportBasis.regular _)] assumption -- TODO: show that G•[S] ∈ AlgebraicSubsets V have GS_in_algsubs_V : G•[S] ∈ AlgebraicSubsets V := by unfold AlgebraicSubsets simp constructor · rw [rigidStabilizerBasis_eq_algebraicCentralizerBasis (α := α)] let ⟨S', S'_eq⟩ := (RegularSupportBasis.mem_asSet S).mp S_rsupp_basis rw [<-S'_eq] rw [RigidStabilizerBasis.mem_iff' _ (LocallyMoving.locally_moving _ S'.regular.isOpen S'.nonempty)] sorry · exact GS_ss_V let ⟨g, g_in_U, W, W_in_F, W_rigid, Wconj_eq_GS⟩ := algsubs_ss_algorb GS_in_algsubs_V use g constructor { rw [<-SetLike.mem_coe] rw [U_rigid.toRegularSupportBasis_eq (α := α)] assumption } use W_rigid.toRegularSupportBasis α constructor · apply Ultrafilter.of_le rw [<-RubinFilter.mem_to_action_filter] assumption · rw [<-rigidStabilizer_eq_iff G] swap { rw [<-smulImage_regular (G := G)] apply RegularSupportBasis.regular } swap { let ⟨S', S'_eq⟩ := (RegularSupportBasis.mem_asSet S).mp S_rsupp_basis rw [<-S'_eq] exact S'.regular } ext i rw [rigidStabilizer_smulImage, <-Wconj_eq_GS, <-IsRigidSubgroup.mem_subgroup, <-SetLike.mem_coe, IsRigidSubgroup.toRegularSupportBasis_eq] simp constructor · intro gig_in_W use g⁻¹ * i * g constructor; exact gig_in_W group · intro ⟨j, j_in_W, gjg_eq_i⟩ rw [<-gjg_eq_i] group assumption end Equivalence -- TODO: prove that for every rubinfilter, there exists an associated ultrafilter on α that converges instance RubinFilterSetoid (G : Type _) [Group G] : Setoid (RubinFilter G) where r F F' := ∀ (U : Set G), IsRigidSubgroup U → ((∃ V : Set G, V ≤ U ∧ AlgebraicSubsets V ⊆ AlgebraicOrbit F U) ↔ (∃ V' : Set G, V' ≤ U ∧ AlgebraicSubsets V' ⊆ AlgebraicOrbit F' U)) iseqv := by constructor · intros simp · intro F F' h intro U U_rigid symm exact h U U_rigid · intro F₁ F₂ F₃ intro h₁₂ h₂₃ intro U U_rigid specialize h₁₂ U U_rigid specialize h₂₃ U U_rigid exact Iff.trans h₁₂ h₂₃ def RubinSpace (G : Type _) [Group G] := Quotient (RubinFilterSetoid G) -- Topology can be generated from the disconnectedness of the filters variable {β : Type _} variable [TopologicalSpace β] [MulAction G β] [ContinuousMulAction G β] instance : TopologicalSpace (RubinSpace G) := sorry instance : MulAction G (RubinSpace G) := sorry theorem rubin' (hα : RubinAction G α) : EquivariantHomeomorph G α (RubinSpace G) := by sorry theorem rubin (hα : RubinAction G α) (hβ : RubinAction G β) : EquivariantHomeomorph G α β := by -- by composing rubin' hα sorry end Rubin end Rubin