| import OAI.GroupTheory.PolycyclicRecognition.Analysis.NormalizedPairPacking |
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| namespace OAI |
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| /-! # Image paths centered at their actual common starting point -/ |
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| noncomputable section |
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| namespace PolycyclicRecognition |
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| open scoped NNReal |
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| variable {G W : Type*} [Group G] [MetricSpace G] |
| [IsTopologicalGroup G] [IsIsometricSMul G G] |
| [NormedAddCommGroup W] [NormedSpace ℝ W] |
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| def basedImage (F : G → G) (a g : G) : G := (F a)⁻¹ * F g |
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| omit [IsTopologicalGroup G] in |
| theorem basedImage_dist (F : G → G) (a g h : G) : |
| dist (basedImage F a g) (basedImage F a h) = dist (F g) (F h) := |
| dist_mul_left _ _ _ |
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| omit [IsTopologicalGroup G] [IsIsometricSMul G G] [MetricSpace G] [NormedSpace ℝ W] in |
| theorem basedImage_height (π : G →* Multiplicative W) (F : G → G) (a g : G) : |
| (π (basedImage F a g)).toAdd = (π (F g)).toAdd - (π (F a)).toAdd := by |
| simp only [basedImage, map_mul, map_inv, toAdd_mul, toAdd_inv] |
| abel |
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| / |
| relative heights, and multiplies both coarse metric constants by the same |
| Lipschitz constant. -/ |
| theorem basedImage_map_path (F : G → G) {K : ℝ≥0} (hF : LipschitzWith K F) |
| {a g : G} (γ : Path a g) {L κ : ℝ} (hγ : PathMetricBound γ L κ) : |
| ∃ η : Path (1 : G) (basedImage F a g), |
| (∀ t, η t = basedImage F a (γ t)) ∧ PathMetricBound η (K * L) (K * κ) := by |
| have hc : Continuous (basedImage F a) := continuous_const.mul hF.continuous |
| let η : Path (1 : G) (basedImage F a g) := (γ.map hc).cast (by simp [basedImage]) rfl |
| refine ⟨η, (fun _ => rfl), ?_⟩ |
| intro s t |
| change dist (basedImage F a (γ s)) (basedImage F a (γ t)) ≤ _ |
| rw [basedImage_dist] |
| calc |
| _ ≤ (K : ℝ) * dist (γ s) (γ t) := hF.dist_le_mul _ _ |
| _ ≤ (K : ℝ) * (L * |(s : ℝ) - t| + κ) := |
| mul_le_mul_of_nonneg_left (hγ s t) K.coe_nonneg |
| _ = _ := by ring |
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| omit [IsTopologicalGroup G] in |
| / |
| unit-separated points, by the actual lower quasi-isometry inequality. -/ |
| theorem basedImage_separated_card [DecidableEq G] {Λ : ℝ≥0} (p : NormalizedPair G Λ) |
| (hΛ : 0 < (Λ : ℝ)) (a : G) (s : Finset G) |
| (hsep : ∀ x ∈ s, ∀ y ∈ s, x ≠ y → (Λ : ℝ) * ((Λ : ℝ) + 1) < dist x y) : |
| (s.image (basedImage p.val.1 a)).card = s.card ∧ |
| ∀ x ∈ s.image (basedImage p.val.1 a), |
| ∀ y ∈ s.image (basedImage p.val.1 a), x ≠ y → 1 ≤ dist x y := by |
| classical |
| have hd (x y : G) (hx : x ∈ s) (hy : y ∈ s) (hne : x ≠ y) : |
| 1 < dist (basedImage p.val.1 a x) (basedImage p.val.1 a y) := by |
| rw [basedImage_dist] |
| exact normalizedPair_image_dist_gt_one p hΛ (hsep x hx y hy hne) |
| constructor |
| · apply Finset.card_image_iff.mpr |
| intro x hx y hy he |
| by_contra hne |
| have hh := hd x y hx hy hne |
| rw [he, dist_self] at hh |
| linarith |
| · intro x hx y hy hne |
| obtain ⟨u, hu, rfl⟩ := Finset.mem_image.mp hx |
| obtain ⟨v, hv, rfl⟩ := Finset.mem_image.mp hy |
| exact (hd u v hu hv (fun he => hne (congrArg _ he))).le |
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| end PolycyclicRecognition |
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| end |
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| end OAI |
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