| import OAI.Analysis.FreeGroupFactor.InfiniteGeneration | |
| namespace OAI | |
| noncomputable section | |
| open Set Metric Filter | |
| open scoped NNReal Topology BigOperators ContDiff ENNReal | |
| attribute [local instance] Classical.propDecidable | |
| namespace FreeGroupFactorMain | |
| namespace InfiniteGeneration | |
| theorem one_step_none (ε : ℝ) (hε : 0 < ε) : | |
| ∃ α : M ≃⋆ₐ[ℂ] M, | |
| (∀ T, canonicalTrace (α T) = canonicalTrace T) ∧ | |
| (∀ k : ℕ, α (extra (some k) : M) = (extra (some k) : M)) ∧ | |
| ∃ w : FreeGroup (Fin 3), | |
| (∀ i, ‖α (initial i : M)-(initial i : M)‖ < ε) ∧ | |
| twoNorm (α (Generation.word initial w)-(extra none : M)) < ε := by | |
| obtain ⟨α,ht,hfix,w,hA,hw⟩ := FiniteActive.one_step (κ := ℕ) 3 (by rfl) hε | |
| refine ⟨α,ht,hfix,w,hA,?_⟩ | |
| rw [word_initial] | |
| exact hw | |
| theorem generating_tuple : | |
| ∃ U : Tuple, | |
| (∀ g : FreeGroup (Fin 3), canonicalTrace (Generation.word U g) = if g = 1 then 1 else 0) ∧ | |
| ⊤ ≤ (Submodule.span ℂ (Set.range (fun g => groupL2 (Generation.word U g)))).topologicalClosure := by | |
| apply generating_haar_tuple | |
| apply orbit_local_approximation | |
| intro Φ ht s θ hθ | |
| apply orbit_step _ Φ ht s hθ | |
| intro s ε hε | |
| apply finite_absorption _ s ε hε | |
| intro q ε hε | |
| exact one_step_extra_of_none one_step_none q hε | |
| theorem rank_three_to_index : HasNormalTracialIso (FreeGroupRank 3) Group := by | |
| classical | |
| obtain ⟨U,ht,htotal⟩ := generating_tuple | |
| apply normalTracialIso_of_haar_total (Generation.word U) | |
| · intro g | |
| exact (Generation.word_inv U g).symm | |
| · exact ht | |
| · exact htotal | |
| def indexEquivNat : Index ≃ ℕ := by | |
| letI : Denumerable Index := Denumerable.ofEncodableOfInfinite Index | |
| exact Denumerable.eqv Index | |
| end InfiniteGeneration | |
| theorem rankThreeInfinite : HasNormalTracialIso (FreeGroupRank 3) (FreeGroup ℕ) := | |
| InfiniteGeneration.rank_three_to_index.trans | |
| (normalTracialIso_of_groupEquiv (FreeGroup.freeGroupCongr InfiniteGeneration.indexEquivNat)) | |
| end FreeGroupFactorMain | |
| end | |
| end OAI | |