File size: 3,515 Bytes
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 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 | import Mathlib
import OAI.GroupTheory.SimpleAmenable.CentralCovers.AlignedPolygonGeneration
namespace OAI
section
section
open scoped symmDiff
namespace SimpleAmenable
open scoped commutatorElement
open scoped commutatorElement
section AlignedFiniteTables
variable {α G : Type*} [Group G] (f : α → G)
theorem finite_table_lift_in_subgroup_eventually (L : ℕ)
(K : Subgroup (BoundedRelationCover L f)) {T : Type*} [Group T] [Finite T]
(φ : T →* G) (hφ : ∀ s, φ s ∈ K.map (coverMap L f)) :
∃ B : ℕ, ∃ hLB : L ≤ B, ∀ M : ℕ, ∀ hBM : B ≤ M,
∃ ρ : T →* BoundedRelationCover M f,
(coverMap M f).comp ρ = φ ∧
ρ.range ≤ K.map (coverTransition f (le_trans hLB hBM)) := by
choose y hy hyq using hφ
choose w hw using fun s => PresentedGroup.mk_surjective (shortRelations L f) (y s)
have hwp : ∀ s, FreeGroup.lift f (w s) = φ s := by
intro s
rw [← coverMap_mk L f,hw s,hyq s]
obtain ⟨B,hB⟩ := finite_table_lift_eventually f φ w hwp
refine ⟨max L B,le_max_left _ _,fun M hM => ?_⟩
obtain ⟨ρ,hρ,hρw⟩ := hB M (le_trans (le_max_right _ _) hM)
refine ⟨ρ,hρ,?_⟩
rintro z ⟨s,rfl⟩
refine ⟨y s,hy s,?_⟩
rw [← hw s,coverTransition_mk,hρw s]
end AlignedFiniteTables
section SourceAlignedTableCompatibility
variable (a : ℕ) (r : CutRing) (m : ℕ) (hm : 2 ≤ m)
theorem sourceAlignedGroup_map_full (M : ℕ)
(t : Multiplicative (FreeAbelianGroup (Fin m × Fin 2)) →*
BoundedRelationCover M (alternatingGenerator a r m hm))
(ht : (coverMap M (alternatingGenerator a r m hm)).comp t = sourceLatticeMap a r m hm)
(I : Finset (Fin (m+1))) :
(sourceAlignedGroup a r m hm M t I).map
((polygonAlternatingGroup a (m+1)).subtype.comp
(coverMap M (alternatingGenerator a r m hm))) = polygonAlignedGroup a r m hm I := by
rw [sourceAlignedGroup,alignedGroup_map]
have htrans : ((polygonAlternatingGroup a (m+1)).subtype.comp
(coverMap M (alternatingGenerator a r m hm))).comp t =
sourceLatticeFullMap a r m hm := by
rw [MonoidHom.comp_assoc,ht]
rfl
rw [htrans]
rfl
theorem sourceAlignedGroup_transition {L M : ℕ} (hLM : L ≤ M)
(t : Multiplicative (FreeAbelianGroup (Fin m × Fin 2)) →*
BoundedRelationCover L (alternatingGenerator a r m hm)) (I : Finset (Fin (m+1))) :
(sourceAlignedGroup a r m hm L t I).map
(coverTransition (alternatingGenerator a r m hm) hLM) =
sourceAlignedGroup a r m hm M
((coverTransition (alternatingGenerator a r m hm) hLM).comp t) I := by
rw [sourceAlignedGroup,alignedGroup_map]
rfl
theorem fiveTrack_conditional_preimage_aligned
(hr : 0 < ordinary r ∧ ordinary r < 1/2) (M : ℕ)
(t : Multiplicative (FreeAbelianGroup (Fin m × Fin 2)) →*
BoundedRelationCover M (alternatingGenerator a r m hm))
(ht : (coverMap M (alternatingGenerator a r m hm)).comp t = sourceLatticeMap a r m hm)
(ι : Fin 5 ↪ Fin (m+1)) (b : Fin (m+1)) (hb : b ∉ Set.range ι)
(U : polygonAlgebra a) (g : alternatingGroup (Fin 5)) :
∃ y ∈ sourceAlignedGroup a r m hm M t (Finset.univ.map ι),
((coverMap M (alternatingGenerator a r m hm) y).val : polygonFullGroup a (m+1)) =
conditionalHom U (fiveTrackHom ι g) := by
have hh := fiveTrack_polygon_aligned a r m hm hr ι b hb U g
rw [← sourceAlignedGroup_map_full a r m hm M t ht] at hh
exact hh
end SourceAlignedTableCompatibility
end SimpleAmenable
end
end
end OAI
|