| 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 |
|
|
