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namespace OAI
noncomputable section
universe uA uι uκ
open Set Metric Filter
open scoped NNReal Topology BigOperators ContDiff ENNReal
attribute [local instance] Classical.propDecidable
namespace FreeGroupFactorMain.FiniteActive
section
open UnitaryEvaluation
open scoped ENNReal
theorem original_generators_mem {ι : Type uι} {κ : Type uκ} [Fintype ι] [DecidableEq ι] [DecidableEq κ]
(h : ι → FreeGroup ι →₀ ℝ) (l : ℤ →₀ ℂ)
(hS : IsSelfAdjoint (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l))
(X : ℝ → ι → Ambient ι κ) (hU : ∀ t i, X t i ∈ unitary (Ambient ι κ))
(hX₀ : X 0 = fun i => groupUnitary (FreeGroup.of (some (Sum.inl i))))
(hX : ∀ t, HasDerivAt X (PolynomialFlow.field
(laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) h (X t)) t)
(E : StarSubalgebra ℂ (Ambient ι κ)) (hE : IsClosed (E : Set (Ambient ι κ)))
(hC : (generatorUnitary (none : Option (ι ⊕ κ)) : Ambient ι κ) ∈ E)
(t₀ : ℝ) (hXt : ∀ i, X t₀ i ∈ E)
(hpass : ∀ k : κ, (generatorUnitary (some (Sum.inr k) : Option (ι ⊕ κ)) : Ambient ι κ) ∈ E) :
∀ i : Option (ι ⊕ κ), (generatorUnitary i : Ambient ι κ) ∈ E := by
have hXS := PolynomialFlow.solution_mem_subalgebra E hE
(laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) hS
(laurentValue_mem E _ hC l) h X hX hU t₀ hXt
intro i
cases i with
| none => exact hC
| some j =>
cases j with
| inl i =>
change groupUnitary (FreeGroup.of (some (Sum.inl i))) ∈ E
simpa only [hX₀] using hXS 0 i
| inr k => exact hpass k
theorem orbit_total {ι : Type uι} {κ : Type uκ} [Fintype ι] [DecidableEq ι] [DecidableEq κ]
(h : ι → FreeGroup ι →₀ ℝ) (l : ℤ →₀ ℂ)
(hS : IsSelfAdjoint (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l))
(X : ℝ → ι → groupVonNeumann (FreeGroup (Option (ι ⊕ κ))))
(hU : ∀ t j, X t j ∈ unitary (groupVonNeumann (FreeGroup (Option (ι ⊕ κ)))))
(hX₀ : X 0 = fun j => groupUnitary (FreeGroup.of (some (Sum.inl j))))
(hX : ∀ t, HasDerivAt X (PolynomialFlow.field
(laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) h (X t)) t) (t₀ : ℝ) :
⊤ ≤ (Submodule.span ℂ (Set.range (fun g : FreeGroup (Option (ι ⊕ κ)) =>
((wordHom (extendUnitary (fun j => ⟨tuple (X t₀) j,tuple_unitary (X t₀) (hU t₀) j⟩)
(generatorUnitary none)) g : groupVonNeumann (FreeGroup (Option (ι ⊕ κ)))) :
GroupOperator (FreeGroup (Option (ι ⊕ κ)))) deltaOne))).topologicalClosure := by
classical
let U := extendUnitary (fun j => ⟨tuple (X t₀) j,tuple_unitary (X t₀) (hU t₀) j⟩) (generatorUnitary none)
let ρ : FreeGroup (Option (ι ⊕ κ)) →* GroupOperator (FreeGroup (Option (ι ⊕ κ))) :=
(groupVonNeumann _).subtype.toMonoidHom.comp (wordHom U)
let K := (Submodule.span ℂ (Set.range (fun g => ρ g deltaOne))).topologicalClosure
let E := (reducingAlgebra K).comap (groupVonNeumann _).subtype
have hK : IsClosed (K : Set (GroupHilbert (FreeGroup (Option (ι ⊕ κ))))) :=
Submodule.isClosed_topologicalClosure _
have hE : IsClosed (E : Set (groupVonNeumann (FreeGroup (Option (ι ⊕ κ))))) :=
(reducingAlgebra_isClosed K hK).preimage continuous_subtype_val
have hρ (g) : wordHom U g ∈ E := by
constructor
· exact closedSpan_invariant ρ deltaOne g
· have he : star (ρ g) = ρ g⁻¹ := by
change ((star (wordHom U g) : groupVonNeumann (FreeGroup (Option (ι ⊕ κ)))) :
GroupOperator (FreeGroup (Option (ι ⊕ κ)))) = _
rw [← wordHom_inv]
rfl
change ∀ x ∈ K, (star (ρ g)) x ∈ K
rw [he]
exact closedSpan_invariant ρ deltaOne g⁻¹
have hC : (generatorUnitary (none : Option (ι ⊕ κ)) : groupVonNeumann _) ∈ E := by
simpa only [wordHom_of, U, extendUnitary, Option.elim_none] using hρ (FreeGroup.of none)
have hXt : ∀ j, X t₀ j ∈ E := by
intro j
simpa only [wordHom_of, U, extendUnitary, Option.elim_some,tuple,Sum.elim_inl] using hρ (FreeGroup.of (some (Sum.inl j)))
have hpass (k : κ) : (generatorUnitary (some (Sum.inr k) : Option (ι ⊕ κ)) : Ambient ι κ) ∈ E := by
change groupUnitary (FreeGroup.of (some (Sum.inr k))) ∈ E
simpa only [wordHom_of,U,extendUnitary,Option.elim_some,tuple,Sum.elim_inr]
using hρ (FreeGroup.of (some (Sum.inr k)))
have hgen := original_generators_mem h l hS X hU hX₀ hX E hE hC t₀ hXt hpass
change ⊤ ≤ K
apply canonical_total_of_reducing K hK _ hgen
apply Submodule.le_topologicalClosure
apply Submodule.subset_span
exact ⟨1, by simp⟩
end
open UnitaryEvaluation
theorem extension_at_time {ι : Type uι} {κ : Type uκ} [Fintype ι] [DecidableEq ι] [DecidableEq κ]
(h : ι → FreeGroup ι →₀ ℝ) (l : ℤ →₀ ℂ)
(hS : IsSelfAdjoint (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l))
(X : ℝ → ι → Ambient ι κ) (hU : ∀ t i, X t i ∈ unitary (Ambient ι κ))
(hX₀ : X 0 = fun i => groupUnitary (FreeGroup.of (some (Sum.inl i))))
(hX : ∀ t, HasDerivAt X (PolynomialFlow.field
(laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) h (X t)) t) (t : ℝ) :
∃ Φ : Ambient ι κ ≃⋆ₐ[ℂ] Ambient ι κ,
(∀ T, canonicalTrace (Φ T) = canonicalTrace T) ∧
@Continuous _ _ (groupUltraweakTopology (FreeGroup (Option (ι ⊕ κ))))
(groupUltraweakTopology (FreeGroup (Option (ι ⊕ κ)))) Φ ∧
∀ g, Φ (groupUnitary g) = wordHom
(extendUnitary (fun j => ⟨tuple (X t) j,tuple_unitary (X t) (hU t) j⟩)
(generatorUnitary none)) g := by
exact iso_extension_of_haar_total
(wordHom (extendUnitary (fun j => ⟨tuple (X t) j,tuple_unitary (X t) (hU t) j⟩)
(generatorUnitary none)))
(fun g => (wordHom_inv _ g).symm)
(moments h l hS X hU hX₀ hX t)
(orbit_total h l hS X hU hX₀ hX t)
theorem basis_solution {ι : Type uι} {κ : Type uκ} [Fintype ι] [DecidableEq ι] [DecidableEq κ]
(h : ι → FreeGroup ι →₀ ℝ) (l : ℤ →₀ ℂ)
(hS : IsSelfAdjoint (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l)) :
∃ (X : ℝ → ι → Ambient ι κ),
X 0 = (fun i => groupUnitary (FreeGroup.of (some (Sum.inl i)))) ∧
(∀ t, HasDerivAt X (PolynomialFlow.field
(laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) h (X t)) t) ∧
∀ t i, X t i ∈ unitary (Ambient ι κ) := by
exact PolynomialFlow.global_unitary_solution (A := Ambient ι κ)
(laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) hS h
(fun i => groupUnitary (FreeGroup.of (some (Sum.inl i)))) (fun _ => groupUnitary_mem_unitary _)
theorem automorphisms {ι : Type uι} {κ : Type uκ} [Fintype ι] [DecidableEq ι] [DecidableEq κ]
(h : ι → FreeGroup ι →₀ ℝ) (l : ℤ →₀ ℂ)
(hS : IsSelfAdjoint (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l)) :
∃ (X : ℝ → ι → Ambient ι κ)
(hU : ∀ t i, X t i ∈ unitary (Ambient ι κ)),
X 0 = (fun i => groupUnitary (FreeGroup.of (some (Sum.inl i)))) ∧
(∀ t, HasDerivAt X (PolynomialFlow.field
(laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) h (X t)) t) ∧
∀ t, ∃ Φ : Ambient ι κ ≃⋆ₐ[ℂ] Ambient ι κ,
(∀ T, canonicalTrace (Φ T) = canonicalTrace T) ∧
@Continuous _ _ (groupUltraweakTopology (FreeGroup (Option (ι ⊕ κ))))
(groupUltraweakTopology (FreeGroup (Option (ι ⊕ κ)))) Φ ∧
∀ g, Φ (groupUnitary g) = wordHom
(extendUnitary (fun j => ⟨tuple (X t) j,tuple_unitary (X t) (hU t) j⟩)
(generatorUnitary none)) g := by
obtain ⟨X,hX₀,hX,hU⟩ := basis_solution (ι := ι) (κ := κ) h l hS
refine ⟨X,hU,hX₀,hX,?_⟩
intro t
exact extension_at_time h l hS X hU hX₀ hX t
end FreeGroupFactorMain.FiniteActive
namespace FreeGroupFactorMain.UnitaryEvaluation
variable {A : Type uA} [CStarAlgebra A] {ι : Type uι} {κ : Type uκ}
theorem hamiltonianLinear_mapDomain (S : A) (U : κ → unitary A)
(f : ι → κ) (k : FreeGroup ι →₀ ℝ) :
hamiltonianLinear S U (Finsupp.mapDomain (FreeGroup.map f) k) =
hamiltonianLinear S (U ∘ f) k := by
classical
simp only [hamiltonianLinear,Finsupp.linearCombination_apply]
rw [Finsupp.sum_mapDomain_index (fun g => by simp)
(fun g a b => by simp [add_smul])]
apply Finsupp.sum_congr
intro g hg
rw [wordHom_map]
end FreeGroupFactorMain.UnitaryEvaluation
namespace FreeGroupFactorMain.FiniteActive
section
open UnitaryEvaluation
variable {ι : Type uι} {κ : Type uκ} [Fintype ι] [DecidableEq ι] [DecidableEq κ]
omit [Fintype ι] in
theorem flow_bound (l : ℤ →₀ ℂ) (h0 : l 0 = 0)
(hS : IsSelfAdjoint (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l))
(K : ℝ) (hK : 0 ≤ K)
(hn : ‖laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l‖ ≤ K)
(U : ι → unitary (Ambient ι κ))
(Φ : Ambient ι κ ≃⋆ₐ[ℂ] Ambient ι κ)
(hΦ : ∀ g, Φ (groupUnitary g) = wordHom
(extendUnitary (fun j => ⟨tuple (fun i => (U i : Ambient ι κ)) j,
tuple_unitary _ (fun i => (U i).property) j⟩) (generatorUnitary none)) g)
(k : FreeGroup ι →₀ ℝ) :
‖hamiltonianLinear (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) U k‖ ≤
3*K*Real.sqrt (k.sum (fun _ r => r^2)) := by
let V : (ι ⊕ κ) → unitary (Ambient ι κ) := fun j =>
⟨tuple (fun i => (U i : Ambient ι κ)) j,tuple_unitary _ (fun i => (U i).property) j⟩
have hb := flow_iso_hamiltonian_bound l h0 hS K hK hn V Φ hΦ
(Finsupp.mapDomain (FreeGroup.map Sum.inl) k)
rw [hamiltonianLinear_mapDomain,
Finsupp.sum_mapDomain_index_inj (FreeGroup.map_injective Sum.inl_injective)] at hb
exact hb
theorem bounded_solution (h : ι → FreeGroup ι →₀ ℝ) (l : ℤ →₀ ℂ) (h0 : l 0 = 0)
(hS : IsSelfAdjoint (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l))
(K : ℝ) (hK : 0 ≤ K)
(hn : ‖laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l‖ ≤ K)
(X : ℝ → ι → Ambient ι κ) (hU : ∀ t i, X t i ∈ unitary (Ambient ι κ))
(hX₀ : X 0 = fun i => groupUnitary (FreeGroup.of (some (Sum.inl i))))
(hX : ∀ t, HasDerivAt X (PolynomialFlow.field
(laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) h (X t)) t) :
∀ t k, ‖hamiltonianLinear
(laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l)
(fun i => ⟨X t i,hU t i⟩) k‖ ≤ 3*K*Real.sqrt (k.sum (fun _ r => r^2)) := by
intro t k
obtain ⟨Φ,_ht,_hc,hΦ⟩ := extension_at_time (ι := ι) (κ := κ) h l hS X hU hX₀ hX t
exact flow_bound (ι := ι) (κ := κ) l h0 hS K hK hn (fun i => ⟨X t i,hU t i⟩) Φ hΦ k
theorem bounded_flow (h : ι → FreeGroup ι →₀ ℝ) (l : ℤ →₀ ℂ) (h0 : l 0 = 0)
(hS : IsSelfAdjoint (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l))
(K : ℝ) (hK : 0 ≤ K)
(hn : ‖laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l‖ ≤ K) :
∃ U : ℝ → ι → unitary (Ambient ι κ),
U 0 = (fun i => generatorUnitary (some (Sum.inl i) : Option (ι ⊕ κ))) ∧
(∀ t, HasDerivAt (fun t i => (U t i : Ambient ι κ))
(PolynomialFlow.field (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l)
h (fun i => (U t i : Ambient ι κ))) t) ∧
(∀ t k, ‖hamiltonianLinear
(laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) (U t) k‖ ≤
3*K*Real.sqrt (k.sum (fun _ r => r^2))) ∧
∃ β : Ambient ι κ ≃⋆ₐ[ℂ] Ambient ι κ,
(∀ T, canonicalTrace (β T) = canonicalTrace T) ∧
@Continuous _ _ (groupUltraweakTopology (FreeGroup (Option (ι ⊕ κ))))
(groupUltraweakTopology (FreeGroup (Option (ι ⊕ κ)))) β ∧
∀ g, β (groupUnitary g) = wordHom
(extendUnitary (fun j => ⟨tuple (fun i => (U 1 i : Ambient ι κ)) j,
tuple_unitary _ (fun i => (U 1 i).property) j⟩) (generatorUnitary none)) g := by
obtain ⟨X,hX₀,hX,hU⟩ := basis_solution (ι := ι) (κ := κ) h l hS
let U (t : ℝ) : ι → unitary (Ambient ι κ) := fun i => ⟨X t i,hU t i⟩
refine ⟨U,?_,hX,?_,?_⟩
· funext i
exact Subtype.ext (congrFun hX₀ i)
· exact bounded_solution (ι := ι) (κ := κ) h l h0 hS K hK hn X hU hX₀ hX
· exact extension_at_time (ι := ι) (κ := κ) h l hS X hU hX₀ hX 1
omit [Fintype ι] [DecidableEq ι] [DecidableEq κ] in
theorem wordHom_active (w : FreeGroup ι) :
wordHom (fun i => generatorUnitary (some (Sum.inl i) : Option (ι ⊕ κ))) w =
groupUnitary (FreeGroup.map (fun i => some (Sum.inl i)) w) := by
rw [← wordHom_generatorUnitary,wordHom_map]
rfl
omit [Fintype ι] [DecidableEq ι] [DecidableEq κ] in
theorem wordHom_tuple_active (U : ι → unitary (Ambient ι κ)) (w : FreeGroup ι) :
wordHom (extendUnitary (fun j => ⟨tuple (fun i => (U i : Ambient ι κ)) j,
tuple_unitary _ (fun i => (U i).property) j⟩) (generatorUnitary none))
(FreeGroup.map (fun i => some (Sum.inl i)) w) = wordHom U w := by
rw [wordHom_map]
rfl
omit [DecidableEq κ] in
theorem displacements_from_bounds
(h : ι → FreeGroup ι →₀ ℝ) (l : ℤ →₀ ℂ)
(hS : IsSelfAdjoint (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l)) (K : ℝ)
(U : ℝ → ι → unitary (Ambient ι κ))
(hU₀ : U 0 = fun j => generatorUnitary (some (Sum.inl j) : Option (ι ⊕ κ)))
(hX : ∀ t, HasDerivAt (fun t j => (U t j : Ambient ι κ))
(PolynomialFlow.field (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) h
(fun j => (U t j : Ambient ι κ))) t)
(hbound : ∀ t k, ‖hamiltonianLinear (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) (U t) k‖ ≤
3*K*Real.sqrt (k.sum (fun _ r => r^2))) :
(∀ j, ‖(U 1 j : Ambient ι κ) - groupUnitary (FreeGroup.of (some (Sum.inl j)))‖ ≤
3*K*Real.sqrt ((h j).sum (fun _ r => r^2))) ∧
∀ w : FreeGroup ι, ‖wordHom (U 1) w - NormedSpace.exp (Complex.I • laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) *
groupUnitary (FreeGroup.map (fun i => some (Sum.inl i)) w)‖ ≤
3*K*Real.sqrt ((freeCocycle h w - Finsupp.single 1 1).sum (fun _ r => r^2)) := by
constructor
· intro j
have hd := generator_displacement (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) h U hX
(fun j => 3*K*Real.sqrt ((h j).sum (fun _ r => r^2)))
(fun t j => hbound t (h j)) j
rw [hU₀] at hd
exact hd
· intro w
have hd := word_displacement (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) hS h U hX w
(3*K*Real.sqrt ((freeCocycle h w - Finsupp.single 1 1).sum (fun _ r => r^2)))
(fun t => hbound t (freeCocycle h w - Finsupp.single 1 1))
rwa [hU₀, wordHom_active] at hd
theorem displacement (h : ι → FreeGroup ι →₀ ℝ) (l : ℤ →₀ ℂ) (h0 : l 0 = 0)
(hS : IsSelfAdjoint (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l))
(K : ℝ) (hK : 0 ≤ K)
(hn : ‖laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l‖ ≤ K) :
∃ β : Ambient ι κ ≃⋆ₐ[ℂ] Ambient ι κ,
(∀ T, canonicalTrace (β T) = canonicalTrace T) ∧
@Continuous _ _ (groupUltraweakTopology (FreeGroup (Option (ι ⊕ κ))))
(groupUltraweakTopology (FreeGroup (Option (ι ⊕ κ)))) β ∧
β (groupUnitary (FreeGroup.of none)) = groupUnitary (FreeGroup.of none) ∧
(∀ k : κ, β (groupUnitary (FreeGroup.of (some (Sum.inr k)))) =
groupUnitary (FreeGroup.of (some (Sum.inr k)))) ∧
(∀ i : ι, ‖β (groupUnitary (FreeGroup.of (some (Sum.inl i) : Option (ι ⊕ κ)))) -
groupUnitary (FreeGroup.of (some (Sum.inl i) : Option (ι ⊕ κ)))‖ ≤
3*K*Real.sqrt ((h i).sum (fun _ r => r^2))) ∧
∀ w : FreeGroup ι, ‖β (groupUnitary (FreeGroup.map (fun i => some (Sum.inl i)) w)) -
NormedSpace.exp (Complex.I • laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) *
groupUnitary (FreeGroup.map (fun i => some (Sum.inl i)) w)‖ ≤
3*K*Real.sqrt ((freeCocycle h w - Finsupp.single 1 1).sum (fun _ r => r^2)) := by
obtain ⟨U,hU₀,hX,hbound,β,hβt,hβc,hβ⟩ := bounded_flow (ι := ι) (κ := κ) h l h0 hS K hK hn
obtain ⟨hgen,hword⟩ := displacements_from_bounds (ι := ι) (κ := κ) h l hS K U hU₀ hX hbound
refine ⟨β,hβt,hβc,?_,?_,?_,?_⟩
· exact (hβ (FreeGroup.of (none : Option (ι ⊕ κ)))).trans (wordHom_of _ _)
· intro k
exact (hβ (FreeGroup.of (some (Sum.inr k) : Option (ι ⊕ κ)))).trans (wordHom_of _ _)
· intro i
rw [hβ]
simpa only [wordHom_of,extendUnitary,Option.elim_some,tuple,Sum.elim_inl] using hgen i
· intro w
rw [hβ]
simpa only [wordHom_tuple_active] using hword w
end
open UnitaryEvaluation
theorem one_step_from_laurent {κ : Type uκ} [DecidableEq κ]
(n : ℕ) {ε : ℝ} (hε : 0 < ε)
(l : ℤ →₀ ℂ) (hl0 : l 0 = 0)
(hlS : IsSelfAdjoint (laurentValue (generatorUnitary (none : Option (Fin n ⊕ κ))) l))
(hln : ‖laurentValue (generatorUnitary (none : Option (Fin n ⊕ κ))) l‖ ≤ 2*(Real.pi+1))
(z : unitary ℂ) (hze : twoNorm (NormedSpace.exp (Complex.I • laurentValue
(generatorUnitary (none : Option (Fin n ⊕ κ))) l) -
(z : ℂ) • groupUnitary (FreeGroup.of none)) < ε/2)
(w : FreeGroup (Fin n)) (h : Coeff (FreeGroup (Fin n)))
(hh : coefficientNorm h < (ε/2)/(3*(2*(Real.pi+1))))
(hw : coefficientNorm (freeCocycle (firstCoefficients h) w-Finsupp.single 1 1) <
(ε/2)/(3*(2*(Real.pi+1)))) :
∃ α : Ambient (Fin n) κ ≃⋆ₐ[ℂ] Ambient (Fin n) κ,
(∀ T, canonicalTrace (α T) = canonicalTrace T) ∧
(∀ k : κ, α (groupUnitary (FreeGroup.of (some (Sum.inr k)))) =
groupUnitary (FreeGroup.of (some (Sum.inr k)))) ∧
∃ w : FreeGroup (Fin n),
(∀ i : Fin n, ‖α (groupUnitary (FreeGroup.of (some (Sum.inl i)))) -
groupUnitary (FreeGroup.of (some (Sum.inl i)))‖ < ε) ∧
twoNorm (α (groupUnitary (FreeGroup.map (fun i => some (Sum.inl i)) w)) -
groupUnitary (FreeGroup.of none)) < ε := by
let K : ℝ := 2*(Real.pi+1)
have hK : 0 < K := by dsimp [K]; positivity
obtain ⟨β,hβt,_hβn,_hβC,hβpass,hβgen,hβword⟩ :=
displacement (κ := κ) (firstCoefficients h) l hl0 hlS K hK.le hln
obtain ⟨γ,hγt,_hγn,_hγi,hγgen,hγC⟩ :=
gauged_nielsen_iso (ι := Fin n ⊕ κ) (FreeGroup.map Sum.inl w) z
have hmap : FreeGroup.map (Option.some : Fin n ⊕ κ → Option (Fin n ⊕ κ))
(FreeGroup.map Sum.inl w) = FreeGroup.map (fun i => some (Sum.inl i)) w := by
have he : (FreeGroup.map (Option.some : Fin n ⊕ κ → Option (Fin n ⊕ κ))).comp
(FreeGroup.map Sum.inl) = FreeGroup.map (fun i => some (Sum.inl i)) := by
apply FreeGroup.ext_hom
intro i
simp
exact DFunLike.congr_fun he w
rw [hmap] at hγC
refine ⟨β.trans γ.symm,?_,?_,w,?_,?_⟩
· intro T
change canonicalTrace (γ.symm (β T)) = canonicalTrace T
rw [trace_symm γ hγt,hβt]
· intro k
change γ.symm (β (groupUnitary (FreeGroup.of (some (Sum.inr k))))) = _
rw [hβpass,← hγgen (Sum.inr k),γ.symm_apply_apply]
exact (hγgen (Sum.inr k)).symm
· intro i
exact one_step_generator_estimate β γ _ (hγgen (Sum.inl i)) hK hε (hβgen i)
((firstCoefficients_norm_le h i).trans_lt hh)
· exact one_step_word_estimate β γ hγt
(generatorUnitary (none : Option (Fin n ⊕ κ)))
(groupUnitaryUHom (FreeGroup (Option (Fin n ⊕ κ)))
(FreeGroup.map (fun i => some (Sum.inl i)) w))
_ z hγC hK (hβword w) hw hze
theorem one_step_of_coefficient_estimate {κ : Type uκ} [DecidableEq κ] (n : ℕ)
(hc : ∀ η : ℝ, 0 < η → ∃ (w : FreeGroup (Fin n)) (h : Coeff (FreeGroup (Fin n))),
coefficientNorm h < η ∧ coefficientNorm (freeCocycle (firstCoefficients h) w-Finsupp.single 1 1) < η)
{ε : ℝ} (hε : 0 < ε) :
∃ α : Ambient (Fin n) κ ≃⋆ₐ[ℂ] Ambient (Fin n) κ,
(∀ T, canonicalTrace (α T) = canonicalTrace T) ∧
(∀ k : κ, α (groupUnitary (FreeGroup.of (some (Sum.inr k)))) =
groupUnitary (FreeGroup.of (some (Sum.inr k)))) ∧
∃ w : FreeGroup (Fin n),
(∀ i : Fin n, ‖α (groupUnitary (FreeGroup.of (some (Sum.inl i)))) -
groupUnitary (FreeGroup.of (some (Sum.inl i)))‖ < ε) ∧
twoNorm (α (groupUnitary (FreeGroup.map (fun i => some (Sum.inl i)) w)) -
groupUnitary (FreeGroup.of none)) < ε := by
obtain ⟨l,hl0,hlS,hln,z,hze⟩ := haar_centered_laurent
(generatorUnitary (none : Option (Fin n ⊕ κ))) (generator_haar _) (half_pos hε)
rw [← laurentValue_eq_value] at hlS hln hze
obtain ⟨w,h,hh,hw⟩ := hc ((ε/2)/(3*(2*(Real.pi+1)))) (by positivity)
exact one_step_from_laurent n hε l hl0 hlS hln z hze w h hh hw
theorem one_step {κ : Type uκ} [DecidableEq κ] (n : ℕ) (hn : 3 ≤ n)
{ε : ℝ} (hε : 0 < ε) :
∃ α : Ambient (Fin n) κ ≃⋆ₐ[ℂ] Ambient (Fin n) κ,
(∀ T, canonicalTrace (α T) = canonicalTrace T) ∧
(∀ k : κ, α (groupUnitary (FreeGroup.of (some (Sum.inr k)))) =
groupUnitary (FreeGroup.of (some (Sum.inr k)))) ∧
∃ w : FreeGroup (Fin n),
(∀ i : Fin n, ‖α (groupUnitary (FreeGroup.of (some (Sum.inl i)))) -
groupUnitary (FreeGroup.of (some (Sum.inl i)))‖ < ε) ∧
twoNorm (α (groupUnitary (FreeGroup.map (fun i => some (Sum.inl i)) w)) -
groupUnitary (FreeGroup.of none)) < ε := one_step_of_coefficient_estimate n (smallCocycle n hn) hε
end FreeGroupFactorMain.FiniteActive
end
end OAI
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