| import OAI.GroupTheory.PolycyclicRecognition.Geometry.IntrinsicContractionModel |
| import OAI.GroupTheory.PolycyclicRecognition.Geometry.IntrinsicCartanHeightAssembly |
| import OAI.GroupTheory.PolycyclicRecognition.Analysis.IntrinsicTraceFromHaar |
| import OAI.GroupTheory.PolycyclicRecognition.Algebra.PublishedRecognitionInputs |
| import OAI.GroupTheory.PolycyclicRecognition.Geometry.CoarsePairHeight |
| import OAI.GroupTheory.PolycyclicRecognition.Analysis.CoarsePairMetricChange |
| import OAI.GroupTheory.PolycyclicRecognition.Geometry.HeightChangeOfCoordinates |
| import OAI.GroupTheory.PolycyclicRecognition.Analysis.SelfQuasiIsometry |
| import Mathlib.Algebra.Lie.CartanExists |
|
|
| namespace OAI |
|
|
| /-! # Uniform bounded height in a fixed metric and an arbitrary norm |
|
|
| The auxiliary Riemannian metric is used only to construct smoothing. The |
| identity comparison transfers every uniformly controlled family from the |
| specified proper chain metric. A bounded linear change of coordinates then |
| returns the estimate to the specified norm on the canonical height space. |
| -/ |
|
|
| noncomputable section |
| namespace PolycyclicRecognition |
|
|
| open Module LieAlgebra MeasureTheory |
| open scoped Manifold NNReal |
|
|
| attribute [local instance] stationaryL2Action zeroMeanL2Action |
|
|
| / |
| def ModelIsUnimodular (R : SimplyConnectedSolvableLieModel) : Prop := |
| letI : IsTopologicalGroup R.carrier := |
| topologicalGroup_of_lieGroup (𝓘(ℝ, R.modelSpace)) ⊤ |
| letI : LocallyCompactSpace R.carrier := |
| Manifold.locallyCompact_of_finiteDimensional (𝓘(ℝ, R.modelSpace)) |
| letI : MeasurableSpace R.carrier := borel _ |
| letI : BorelSpace R.carrier := ⟨rfl⟩ |
| (Measure.haar : Measure R.carrier).IsMulRightInvariant |
|
|
| / |
| fixed metric can be any left-invariant proper Riemannian metric. The finite |
| group is chosen before the quasi-isometry constants, and the bound is |
| uniform over all self quasi-isometries with those constants. -/ |
| theorem uniform_bounded_height |
| (R : SimplyConnectedSolvableLieModel) (d : ProperChainMetric R.carrier) |
| {a : ℕ} (T : TriangularAdjoint (GroupLieAlgebra (𝓘(ℝ, R.modelSpace)) R.carrier) a) |
| (huni : ModelIsUnimodular R) |
| {W : Type} [NormedAddCommGroup W] [NormedSpace ℝ W] [FiniteDimensional ℝ W] |
| (q : HeightSpace T.character ≃ₗ[ℝ] W) |
| (Exp : GroupLieAlgebra (𝓘(ℝ, R.modelSpace)) R.carrier → R.carrier) |
| (hExp : IsGroupExponential Exp) (π : R.carrier →* Multiplicative W) |
| (hπExp : ∀ x, (π (Exp x)).toAdd = q (heightProjection T.character x)) : |
| letI := d.compatibleMetric |
| ∃ A : Subgroup (W ≃ₗ[ℝ] W), Set.Finite (A : Set (W ≃ₗ[ℝ] W)) ∧ |
| ∀ K C : ℝ, 1 ≤ K → 0 ≤ C → ∃ B : ℝ, 0 ≤ B ∧ |
| ∀ F : SelfQuasiIsometry R.carrier K C, ∃ AF : A, ∀ g, |
| ‖(π (F.toFun g)).toAdd - ((π (F.toFun 1)).toAdd + AF.val (π g).toAdd)‖ ≤ B := by |
| classical |
| let : IsTopologicalGroup R.carrier := |
| topologicalGroup_of_lieGroup (𝓘(ℝ, R.modelSpace)) ⊤ |
| let : LocallyCompactSpace R.carrier := |
| Manifold.locallyCompact_of_finiteDimensional (𝓘(ℝ, R.modelSpace)) |
| let : MeasurableSpace R.carrier := borel _ |
| let : BorelSpace R.carrier := ⟨rfl⟩ |
| let m : Measure R.carrier := Measure.haar |
| let : m.IsMulRightInvariant := huni |
| obtain ⟨d₀, hcompat⟩ := lieRiemannianMetricInput R |
| let := d₀.toProperChainMetric.compatibleMetric |
| let := d₀.toProperChainMetric.proper_compatibleMetric |
| let := d₀.toProperChainMetric.isometric_compatibleMetric |
| let V := EuclideanSpace ℝ (Fin (finrank ℝ (HeightSpace T.character))) |
| let q₀ : HeightSpace T.character ≃ₗ[ℝ] V := |
| LinearEquiv.ofFinrankEq _ _ (by simp [V]) |
| obtain ⟨π₀, hπ₀Exp, _, _, L, hL, hπ₀Lip⟩ := |
| exists_lipschitz_intrinsic_canonical_height T q₀ d₀.toProperChainMetric.chains_compatibleMetric hExp |
| let e : V ≃ₗ[ℝ] W := q₀.symm.trans q |
| have hπe (g : R.carrier) : (π g).toAdd = e (π₀ g).toAdd := by |
| apply height_eq_of_exponential_linear hExp π π₀ e.toLinearMap |
| intro x |
| rw [hπExp, hπ₀Exp] |
| simp [e] |
| have hπ₀ : LipschitzWith (⟨L, hL⟩ : ℝ≥0) (fun g => (π₀ g).toAdd) := by |
| apply LipschitzWith.of_dist_le_mul |
| intro x y |
| rw [dist_eq_norm] |
| exact hπ₀Lip x y |
| obtain ⟨n, φ, hφ0, hf, hb⟩ := |
| exists_intrinsic_smooth_coordinates T q₀ Exp hExp π₀ hπ₀.continuous hπ₀Exp |
| obtain ⟨hfLip, hbLip⟩ := hcompat n φ hf hb |
| let c := ControlledContraction.ofChart φ hφ0 hfLip hbLip |
| have hunimod (x : GroupLieAlgebra (𝓘(ℝ, R.modelSpace)) R.carrier) : |
| LinearMap.trace ℝ _ (ad ℝ _ x) = 0 := |
| trace_ad_eq_zero_of_intrinsic_kernel_haar T q₀ Exp hExp π₀ hπ₀.continuous hπ₀Exp m x |
| obtain ⟨x, hx⟩ := LieAlgebra.exists_isCartanSubalgebra_engel ℝ |
| (GroupLieAlgebra (𝓘(ℝ, R.modelSpace)) R.carrier) |
| let H := LieSubalgebra.engel ℝ x |
| let : H.IsCartanSubalgebra := hx |
| obtain ⟨M⟩ := nonempty_cartanWordModel H T |
| let D := M.supportIn T hunimod q₀ |
| have hheight := M.exists_uniform_height_intrinsic T hunimod q₀ Exp hExp |
| m π₀ hπ₀Exp hπ₀ |
| d₀.minimizing_compatibleMetric remezPolynomialInput geodesicMetricExistenceInput |
| let A := (templateSymmetryGroup D).map (heightConjugation e) |
| refine ⟨A, finite_heightConjugation e (templateSymmetryGroup D), ?_⟩ |
| intro K C hK hC |
| let Q := @SelfQuasiIsometry R.carrier d.compatibleMetric.toPseudoMetricSpace K C |
| let f : Q → R.carrier → R.carrier := by |
| let := d.compatibleMetric |
| exact fun F => F.toFun |
| let g : Q → R.carrier → R.carrier := by |
| let := d.compatibleMetric |
| exact fun F => F.inverse |
| have hK₀ : 0 ≤ K := le_trans zero_le_one hK |
| have hf : letI := d.compatibleMetric |
| ∀ F x y, dist (f F x) (f F y) ≤ K * dist x y + (C + 3 * K * C) := by |
| let := d.compatibleMetric |
| intro F x y |
| exact (F.upper x y).trans (by nlinarith) |
| have hg : letI := d.compatibleMetric |
| ∀ F x y, dist (g F x) (g F y) ≤ K * dist x y + (C + 3 * K * C) := by |
| let := d.compatibleMetric |
| intro F x y |
| exact (F.inverse_upper hK₀ x y).trans (by linarith) |
| have hgf : letI := d.compatibleMetric |
| ∀ F x, dist (g F (f F x)) x ≤ C + 2 * K * C := by |
| let := d.compatibleMetric |
| intro F x |
| exact (F.left_inverse_close hK₀ x).trans (by linarith) |
| have hfg : letI := d.compatibleMetric |
| ∀ F x, dist (f F (g F x)) x ≤ C + 2 * K * C := by |
| let := d.compatibleMetric |
| intro F x |
| exact ((dist_comm _ _).trans_le (F.inverse_close x)).trans (by nlinarith) |
| obtain ⟨κ, ξ, ε, hκ, hξ, _, hf', hg', hgf', hfg'⟩ := |
| coarse_pair_metric_change d d₀.toProperChainMetric hK₀ |
| (by positivity : 0 ≤ C + 3 * K * C) (by positivity : 0 ≤ C + 2 * K * C) |
| f g hf hg hgf hfg |
| obtain ⟨label, B, hB, hbound⟩ := uniform_affine_height_of_coarse_inverse_pairs |
| c d₀.minimizing_compatibleMetric D π₀ hL hπ₀Lip hheight |
| hκ hξ f g hf' hg' hgf' hfg' |
| refine ⟨‖e.toContinuousLinearEquiv.toContinuousLinearMap‖ * B, |
| mul_nonneg (norm_nonneg _) hB, ?_⟩ |
| intro F |
| refine ⟨⟨heightConjugation e (label F).val, ?_⟩, fun y => ?_⟩ |
| · exact ⟨(label F).val, (label F).property, rfl⟩ |
| · simp only [hπe] |
| exact norm_height_difference_le e _ _ _ (label F).val (hbound F y) |
|
|
| / |
| is no longer needed. -/ |
| theorem uniform_bounded_height_of_published_inputs |
| (_inputs : PublishedRecognitionInputs) |
| (R : SimplyConnectedSolvableLieModel) (d : ProperChainMetric R.carrier) |
| {a : ℕ} (T : TriangularAdjoint (GroupLieAlgebra (𝓘(ℝ, R.modelSpace)) R.carrier) a) |
| (huni : ModelIsUnimodular R) |
| {W : Type} [NormedAddCommGroup W] [NormedSpace ℝ W] [FiniteDimensional ℝ W] |
| (q : HeightSpace T.character ≃ₗ[ℝ] W) |
| (Exp : GroupLieAlgebra (𝓘(ℝ, R.modelSpace)) R.carrier → R.carrier) |
| (hExp : IsGroupExponential Exp) (π : R.carrier →* Multiplicative W) |
| (hπExp : ∀ x, (π (Exp x)).toAdd = q (heightProjection T.character x)) : |
| letI := d.compatibleMetric |
| ∃ A : Subgroup (W ≃ₗ[ℝ] W), Set.Finite (A : Set (W ≃ₗ[ℝ] W)) ∧ |
| ∀ K C : ℝ, 1 ≤ K → 0 ≤ C → ∃ B : ℝ, 0 ≤ B ∧ |
| ∀ F : SelfQuasiIsometry R.carrier K C, ∃ AF : A, ∀ g, |
| ‖(π (F.toFun g)).toAdd - ((π (F.toFun 1)).toAdd + AF.val (π g).toAdd)‖ ≤ B := by |
| exact uniform_bounded_height R d T huni q Exp hExp π hπExp |
|
|
| end PolycyclicRecognition |
|
|
| end |
|
|
| end OAI |
|
|
