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/-
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 : ιι') {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
variable {G α β : Type _} [Group G]
----------------------------------------------------------------
section RubinActions
variable [TopologicalSpace α] [TopologicalSpace β]
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
-- 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 : ) • 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 .dvd yet, nor .mod, nor Int.natAbs, nor Int.div, etc.
have h: (12 : ) = (12 : ) := 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 : ) • 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^(ji)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 (RigidStabilizer 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 : ) •'' V) (g ^ (j : ) •'' 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 : ) > 0 → g ^ (i : ) • 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 : ) • q = g ^ (i : ) • 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 : ) := 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 : ) → (h*g)^(i : ) • 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 > 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 = RigidStabilizer 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} :
RigidStabilizer G (RegularSupport α f) ⊓ RigidStabilizer 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
lemma rigidStabilizerInter_eq_algebraicCentralizerInter {S : Finset G} :
RigidStabilizerInter₀ α S = AlgebraicCentralizerInter₀ S :=
by
unfold RigidStabilizerInter₀
unfold AlgebraicCentralizerInter₀
conv => {
lhs
congr; intro; congr; intro
rw [<-proposition_2_1]
}
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]
#check RegularSupportBasis.asSet
#check RigidStabilizerBasis
-- TODO: implement Smul of G on RigidStabilizerBasis?
-- theorem regularSupportBasis_eq_ridigStabilizerBasis :
-- RegularSupportBasis.asSet α = RigidStabilizerBasis (HomeoGroup α) α :=
-- by
-- sorry
-- TODO: implement Membership on RegularSupportBasis
-- TODO: wrap these things in some neat structures
-- theorem proposition_3_5 {G α : Type _} [Group G] [TopologicalSpace α] [MulAction G α]
-- [T2Space α] [LocallyCompactSpace α] [h_ld : LocallyDense G α] [HasNoIsolatedPoints α]
-- [hc : ContinuousMulAction G α]
-- (U : RegularSupportBasis α) (F: Filter α):
-- (∃ p ∈ U.val, F.HasBasis (fun S: Set α => S ∈ RegularSupportBasis.asSet α ∧ p ∈ S) id)
-- ↔ ∃ V : RegularSupportBasis α, V ≤ U ∧ {W : RegularSupportBasis α | W ≤ V} ⊆ { g •'' W | (g ∈ RigidStabilizer G U.val) (W ∈ F) (_: W ∈ RegularSupportBasis.asSet α) }
-- :=
-- by
-- constructor
-- {
-- simp
-- intro p p_in_U filter_basis
-- have assoc_poset_basis := RegularSupportBasis.isBasis G α
-- have F_eq_nhds : F = 𝓝 p := by
-- have nhds_basis := assoc_poset_basis.nhds_hasBasis (a := p)
-- rw [<-filter_basis.filter_eq]
-- rw [<-nhds_basis.filter_eq]
-- have p_in_int_cl := h_ld.isLocallyDense U U.regular.isOpen p p_in_U
-- -- TODO: show that ∃ V ⊆ closure (orbit (rist G U) p)
-- sorry
-- }
-- sorry
end HomeoGroup
variable [TopologicalSpace α] [MulAction G α] [ContinuousMulAction G α]
[TopologicalSpace β] [MulAction G β] [ContinuousMulAction G β]
theorem rubin (hα : RubinAction G α) (hβ : RubinAction G β) : EquivariantHomeomorph G α β := by
sorry
end Rubin
end Rubin