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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 | import Mathlib
import OAI.GroupTheory.SimpleAmenable.Configurations.FiveTrackGeneration
namespace OAI
section
section
open scoped symmDiff
namespace SimpleAmenable
open scoped commutatorElement
open scoped commutatorElement
section ConditionalTranslationStabilizers
variable {a m : ℕ}
theorem trackTranslation_conditional_commute (U : polygonAlgebra a)
(σ : Equiv.Perm (Fin m)) (d : Fin m → CutRing × CutRing)
(hd : ∀ i, σ i ≠ i → d i = 0) :
Commute (trackTranslation d) (conditionalHom U σ) := by
have he := conditional_translation_conjugate U σ d 0 hd
have hU : (⟨translate a (-0) ⁻¹' U.val,polygon_preimage_translate (-0) U.property⟩ :
polygonAlgebra a) = U := by
apply Subtype.ext
ext p
simp only [Set.mem_preimage,neg_zero,translate_zero]
rw [hU] at he
have hx := congrArg (fun z => z * trackTranslation d) he
change _*_ = _*_
simpa only [mul_assoc,inv_mul_cancel,mul_one] using hx.symm
theorem conditional_conjugation_support (U : polygonAlgebra a)
(σ : Equiv.Perm (Fin m)) (d e : Fin m → CutRing × CutRing)
(hde : ∀ i, σ i ≠ i → d i = e i) :
trackTranslation d * conditionalHom U σ * (trackTranslation d)⁻¹ =
trackTranslation e * conditionalHom U σ * (trackTranslation e)⁻¹ := by
have hc := trackTranslation_conditional_commute U σ (d-e) (by
intro i hi
change d i - e i = 0
rw [hde i hi,sub_self])
rw [trackTranslation_sub] at hc
have hcomm : Commute (trackTranslation (a := a) d) (trackTranslation e) := by
change _*_ = _*_
rw [← trackTranslation_add,← trackTranslation_add,add_comm d e]
have he := congrArg (fun z => trackTranslation e * z * (trackTranslation d)⁻¹) hc.eq
rw [hcomm.inv_right.eq] at he
have hh : trackTranslation e * ((trackTranslation e)⁻¹ * trackTranslation d * conditionalHom U σ) *
(trackTranslation d)⁻¹ = trackTranslation d * conditionalHom U σ * (trackTranslation d)⁻¹ := by group
rw [hh] at he
convert he using 1; group
end ConditionalTranslationStabilizers
end SimpleAmenable
end
end
end OAI
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