CartanPathLifts.lean`

1 declarations, 0 contain sorry, 0 declared as axioms

Fixed factor lifts of height-controlled coarse paths

The chosen lift works for every bounded right increment simultaneously. This is the form used for the two coordinate restrictions in tile counting.

theorem OAI.PolycyclicRecognition.CartanWordModel.exists_coarse_path_thickening_bound line 41

Source documentation

Unit subdivision of an actual coarse path supplies a fixed factor lift with polynomial coordinate bounds, including a prescribed compact right thickening. All heights refer to the original homomorphism.

theorem exists_coarse_path_thickening_bound
    (b₀ : Basis (Fin (finrank ℝ (cartanRootIdeal H))) ℝ (cartanRootIdeal H))
    (γ₀ : M.Bases) {U : Set (F.N.carrier × F.D.carrier)} (hU : IsCompact U)
    (hφ : ∀ d, F.action d = M.adjointGroupAutomorphism T F.N.equiv F.ExpN F.isExpN
      (by rw [← show chartN = F.ExpN from funext hchartN]; exact chartN.bijective) dynkinN
      (cartanLog H F.D.equiv F.ExpD
        (by rw [← show chartD = F.ExpD from funext hchartD]; exact chartD.bijective) d))
    (hpath : ∀ g : G, ∃ γ : ℝ → G, γ 0 = 1 ∧ γ 1 = g ∧
      ∀ t ∈ Icc (0 : ℝ) 1, ∀ u ∈ Icc (0 : ℝ) 1,
        dist (γ t) (γ u) ≤ dist 1 g * |t - u|)
    {K : ℝ} (hK : 0 ≤ K)
    (hLip : ∀ x y, ‖(π x).toAdd - (π y).toAdd‖ ≤ K * dist x y) :
…

Mentions: OAI.PolycyclicRecognition.cartanRootIdeal, OAI.PolycyclicRecognition.cartanLog