File size: 2,058 Bytes
67aee1d | 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 | 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
|