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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 | import Mathlib
import OAI.GroupTheory.SimpleAmenable.CentralCovers.InitialCoverSystem
namespace OAI
section
section
open scoped symmDiff
namespace SimpleAmenable
open scoped commutatorElement
open scoped commutatorElement
section CompiledTranslations
noncomputable def latticeWord {m : ℕ}
(k : Multiplicative (FreeAbelianGroup (Fin m × Fin 2))) :
FreeGroup (SourceGeneratorLabel m) :=
FreeGroup.map (sourceTranslationLabel m)
(translationWordMap_surjective k).choose
theorem latticeWord_eval {m : ℕ} {H : Type*} [Group H]
(f : SourceGeneratorLabel m → H)
(t : Multiplicative (FreeAbelianGroup (Fin m × Fin 2)) →* H)
(ht : ∀ i, t (Multiplicative.ofAdd (FreeAbelianGroup.of i)) = f (sourceTranslationLabel m i))
(k : Multiplicative (FreeAbelianGroup (Fin m × Fin 2))) :
FreeGroup.lift f (latticeWord k) = t k := by
have he : (FreeGroup.lift f).comp (FreeGroup.map (sourceTranslationLabel m)) =
t.comp (translationWordMap (Fin m × Fin 2)) := by
ext i
simpa only [MonoidHom.comp_apply,FreeGroup.map.of,FreeGroup.lift_apply_of,
translationWordMap] using (ht i).symm
have hv := DFunLike.congr_fun he (translationWordMap_surjective k).choose
simpa only [MonoidHom.comp_apply,latticeWord,(translationWordMap_surjective k).choose_spec] using hv
namespace InitialCoverSystem
variable {a m M : ℕ} {r : CutRing} {hm : 2 ≤ m}
(B : InitialCoverSystem a r m hm M)
theorem latticeWord_mk (k : Multiplicative (FreeAbelianGroup (Fin m × Fin 2))) :
PresentedGroup.mk (shortRelations M (alternatingGenerator a r m hm)) (latticeWord k) = B.t k := by
have he : FreeGroup.lift (PresentedGroup.of : SourceGeneratorLabel m →
BoundedRelationCover M (alternatingGenerator a r m hm)) =
PresentedGroup.mk (shortRelations M (alternatingGenerator a r m hm)) := by
ext i
rfl
rw [← he]
exact latticeWord_eval _ B.t B.t_base k
end InitialCoverSystem
end CompiledTranslations
end SimpleAmenable
end
end
end OAI
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