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0dd3af4 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 | import Mathlib
import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.SmallControlTransfer
import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.InvariantPatching
namespace OAI
section
section
open scoped symmDiff
namespace SimpleAmenable
open scoped commutatorElement
open scoped commutatorElement
section ControlledPatching
variable {α H ι : Type*} [Fintype α] [DecidableEq α] [Group H]
[Group.IsPerfect (alternatingGroup α)]
theorem controlled_cell_normalizes (L : Finset α → Subgroup H)
(c : alternatingGroup α →* H)
(hc : ∀ σ I, ∀ x ∈ L I, c σ*x*(c σ)⁻¹ ∈ L (I.map σ.val.toEmbedding))
(hd : ∀ I J, Disjoint I J → ∀ x ∈ L I, ∀ y ∈ L J, Commute x y)
(hα : 20 ≤ Fintype.card α)
(F p v : ι → TrackStar α →* H) (j k : TrackStar α →* H)
(hp : ∀ i, SmallSupported L (p i)) (hv : ∀ i, SmallSupported L (v i))
(hj : SmallSupported L j) (hpc : ∀ i, SmallControlled c (p i) j)
(hvc : ∀ i, SmallControlled c (v i) k)
(hjk : ∀ s t, Commute (j s) (k t))
(hsplit : ∀ i s, F i s=p i s*v i s) :
∀ i s, F i s ∈ Subgroup.normalizer ((⨆ i, (p i).range : Subgroup H) : Set H) := by
have hcomm (i i' : ι) (s t : TrackStar α) : Commute (p i s) (v i' t) :=
small_control_disjoint L c hc hd hα _ _ _ _ (hp i) (hv i') hj
(hpc i) (hvc i') hjk s t
have ho (i : ι) (s : TrackStar α) :
v i s ∈ Subgroup.centralizer ((⨆ i, (p i).range : Subgroup H) : Set H) := by
have hle : (⨆ i, (p i).range : Subgroup H) ≤ Subgroup.centralizer ({v i s} : Set H) := by
apply iSup_le
rintro i' x ⟨t,rfl⟩ y hy
obtain rfl := Set.mem_singleton_iff.mp hy
exact (hcomm i' i t s).eq.symm
intro x hx
exact (hle hx _ rfl).symm
intro i s
exact normalizes_of_cell_split _ _ _ _ (hsplit i s)
((le_iSup (fun i => (p i).range) i) ⟨s,rfl⟩) (ho i s)
theorem sameActionOn_iSup {D : Type*} (f g : D → H) (P : ι → Subgroup H)
(h : ∀ i, SameActionOn f g (P i)) : SameActionOn f g (⨆ i, P i) := by
intro d x hx
let F : H →* H := (MulAut.conj (f d)).toMonoidHom
let J : H →* H := (MulAut.conj (g d)).toMonoidHom
have hle : (⨆ i, P i) ≤ F.eqLocus J := by
apply iSup_le
intro i y hy
exact h i d y hy
exact hle hx
end ControlledPatching
end SimpleAmenable
end
end
end OAI
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