File size: 3,445 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 | import Mathlib
import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.OffsetFrames
namespace OAI
section
section
open scoped symmDiff
namespace SimpleAmenable
open scoped commutatorElement
open scoped commutatorElement
section FrameCompatibility
namespace OffsetFrame
variable {a m : ℕ} {r : CutRing} {hm : 2 ≤ m} {I : Finset (Fin (m+1))}
{u : Fin (m+1) → CutRing × CutRing}
def shift (F : OffsetFrame a r m hm I u)
(k : Multiplicative (FreeAbelianGroup (Fin m × Fin 2)))
(d : Fin (m+1) → CutRing × CutRing)
(hd : sourceLatticeFullMap a r m hm k = trackTranslation d) :
OffsetFrame a r m hm I (d+u) where
k := k*F.k
d := d+F.d
projection := by rw [map_mul,hd,F.projection,trackTranslation_add]
prescribed := by intro i hi; simp only [Pi.add_apply,F.prescribed i hi]
end OffsetFrame
namespace InitialCoverSystem
variable {a m M : ℕ} {r : CutRing} {hm : 2 ≤ m}
(B : InitialCoverSystem a r m hm M)
[Group.IsPerfect (alternatingGroup (Fin (m+1)))]
(hlarge : 15 < m+1) (h : B.AllPrimitiveLaws) (hr : 0<ordinary r ∧ ordinary r<1/2)
theorem frameStar_shift (I : Finset (Fin (m+1))) [Group.IsPerfect (alternatingGroup I)]
(u : Fin (m+1) → CutRing × CutRing) (F : OffsetFrame a r m hm I u)
(k : Multiplicative (FreeAbelianGroup (Fin m × Fin 2)))
(d : Fin (m+1) → CutRing × CutRing)
(hd : sourceLatticeFullMap a r m hm k = trackTranslation d) (V : polygonAlgebra a) :
B.frameStar hlarge h hr I (d+u) (F.shift k d hd) V =
(MulAut.conj (B.t k)).toMonoidHom.comp (B.frameStar hlarge h hr I u F V) := by
ext s : 1
change B.t (k*F.k) * B.polygonStar hlarge h hr V (universalMap (subtypeAlternatingHom I) s) *
(B.t (k*F.k))⁻¹ = B.t k *
(B.t F.k * B.polygonStar hlarge h hr V (universalMap (subtypeAlternatingHom I) s) * (B.t F.k)⁻¹) * (B.t k)⁻¹
simp only [map_mul,mul_inv_rev,mul_assoc]
theorem frameStar_reparameterize (I : Finset (Fin (m+1))) (hI : 5 ≤ I.card)
[Group.IsPerfect (alternatingGroup I)] (b : Fin (m+1)) (hb : b ∉ I)
(u : Fin (m+1) → CutRing × CutRing) (v : CutRing × CutRing)
(F : OffsetFrame a r m hm I u)
(G : OffsetFrame a r m hm I (fun i => u i-v)) (V : polygonAlgebra a) :
B.frameStar hlarge h hr I u F V =
B.frameStar hlarge h hr I (fun i => u i-v) G (spatialTranslate v V) := by
let T := commonFrame a r m hm I b hb v
let K : OffsetFrame a r m hm I u := {
k := G.k*T.k
d := G.d+T.d
projection := by rw [map_mul,G.projection,T.projection,trackTranslation_add]
prescribed := by
intro i hi
simp only [Pi.add_apply,G.prescribed i hi,T.common i hi,sub_add_cancel] }
rw [B.frameStar_independent hlarge h hr I hI b hb u u F K (fun _ _ => rfl) V]
ext s : 1
have he := DFunLike.congr_fun (B.polygonStar_common_translate hlarge h hr I hI b hb v T V) s
change B.t T.k * B.polygonStar hlarge h hr V (universalMap (subtypeAlternatingHom I) s) * (B.t T.k)⁻¹ =
B.polygonStar hlarge h hr (spatialTranslate v V) (universalMap (subtypeAlternatingHom I) s) at he
change B.t (G.k*T.k) * B.polygonStar hlarge h hr V (universalMap (subtypeAlternatingHom I) s) *
(B.t (G.k*T.k))⁻¹ = B.t G.k *
B.polygonStar hlarge h hr (spatialTranslate v V) (universalMap (subtypeAlternatingHom I) s) * (B.t G.k)⁻¹
rw [← he]
simp only [map_mul,mul_inv_rev,mul_assoc]
end InitialCoverSystem
end FrameCompatibility
end SimpleAmenable
end
end
end OAI
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