| import OAI.GroupTheory.Thompson.StandardCharacterization |
| import OAI.GroupTheory.Thompson.FiniteBoundary |
| import OAI.GroupTheory.Thompson.AnalyticConstruction |
| import OAI.GroupTheory.Thompson.MeanToFolner |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open scoped symmDiff |
|
|
| namespace ThompsonNonamenability |
|
|
| local instance : DecidableEq F := Classical.decEq F |
|
|
| theorem thompson_F_boundary_with_exact_constant : |
| ∃ (D : ℕ) (S : Finset F), |
| 2 ≤ D ∧ |
| 4 * Analytic.displacementLipschitzConstant ^ 2 / (1 / 2 : ℝ) ^ 2 < |
| (D : ℝ) ∧ |
| S.Nonempty ∧ |
| 0 < boundaryConstant Analytic.displacementLipschitzConstant (1 / 2) D ∧ |
| ∀ A : Finset F, A.Nonempty → |
| ∃ h ∈ S, |
| boundaryConstant Analytic.displacementLipschitzConstant (1 / 2) D ≤ |
| boundaryRatio A h := by |
| obtain ⟨D, hD, hlarge⟩ := |
| exists_branching_parameter Analytic.displacementLipschitzConstant (1 / 2) |
| obtain ⟨S, hS, hb, hboundary⟩ := |
| exists_fixed_boundary Analytic.displacementMap |
| Analytic.displacementLipschitzConstant (1 / 2) |
| Analytic.displacementLipschitzConstant_pos (by norm_num) |
| Analytic.displacementMap_lipschitz Analytic.displacementMap_displacement |
| D hD hlarge |
| exact ⟨D, S, hD, hlarge, hS, hb, hboundary⟩ |
|
|
| /-- The displayed formula has no additional normalization or hidden factor. -/ |
| theorem thompson_F_boundary_constant_formula (D : ℕ) : |
| boundaryConstant Analytic.displacementLipschitzConstant (1 / 2) D = |
| ((1 / 2 : ℝ) ^ 2 / Analytic.displacementLipschitzConstant ^ 2 - |
| 4 / (D : ℝ)) / (4 * (1 - 1 / (D : ℝ))) := rfl |
|
|
| /-- A fixed finite subset of the genuine PL group uniformly expands every |
| nonempty finite subset, in the unnormalized symmetric-difference form. -/ |
| theorem thompson_F_uniform_boundary : |
| ∃ (S : Finset F) (b : ℝ), S.Nonempty ∧ 0 < b ∧ |
| ∀ A : Finset F, A.Nonempty → |
| ∃ h ∈ S, b * (A.card : ℝ) ≤ ((leftTranslate h A ∆ A).card : ℝ) := by |
| obtain ⟨D, S, _hD, _hlarge, hS, hb, hboundary⟩ := |
| thompson_F_boundary_with_exact_constant |
| refine ⟨S, boundaryConstant Analytic.displacementLipschitzConstant (1 / 2) D, |
| hS, hb, ?_⟩ |
| intro A hA |
| obtain ⟨h, hh, hratio⟩ := hboundary A hA |
| have hcard : (0 : ℝ) < (A.card : ℝ) := |
| Nat.cast_pos.mpr (Finset.card_pos.mpr hA) |
| refine ⟨h, hh, ?_⟩ |
| change boundaryConstant Analytic.displacementLipschitzConstant (1 / 2) D ≤ |
| ((leftTranslate h A ∆ A).card : ℝ) / (A.card : ℝ) at hratio |
| exact (le_div_iff₀ hcard).mp hratio |
|
|
| /-- Standard Thompson F admits no positive normalized left-invariant mean on |
| all bounded real functions. The carrier is identified with the ordinary |
| finite-breakpoint dyadic PL homeomorphisms by `F.standardPLEquiv`. -/ |
| theorem thompson_F_not_amenable : ¬ Nonempty (InvariantMean F) := by |
| obtain ⟨S, b, _hS, hb, hboundary⟩ := thompson_F_uniform_boundary |
| exact not_amenable_of_uniform_boundary S hb hboundary |
|
|
| theorem thompson_F_nonamenable_composition : |
| ∃ group : Group F, letI := group; |
| (∀ (h g : F) (x : UnitInterval), |
| (h * g).val.toHomeomorph x = h.val.toHomeomorph (g.val.toHomeomorph x)) ∧ |
| ¬ Nonempty (InvariantMean F) := by |
| refine ⟨inferInstance, ?_, thompson_F_not_amenable⟩ |
| intro h g x |
| exact F.mul_apply h g x |
|
|
| end ThompsonNonamenability |
|
|
| end |
|
|
| end OAI |
|
|
