| import Mathlib | |
| namespace OAI | |
| namespace Release075 | |
| universe u v | |
| /-- No nonidentity element has finite order. This does not assert unique roots. -/ | |
| def TorsionFree (G : Type u) [Group G] : Prop := | |
| ∀ (g : G) (n : ℕ), 0 < n → g ^ n = 1 → g = 1 | |
| /-- A shortest edge path in the unit-edge graph. -/ | |
| def Geodesic {V : Type u} (X : SimpleGraph V) {x y : V} (p : X.Walk x y) : Prop := | |
| p.length = X.dist x y | |
| /-- Each vertex of `p` is within `δ` of a vertex of `q` or `r`. -/ | |
| def SideThin {V : Type u} (X : SimpleGraph V) (δ : ℕ) {x y z : V} | |
| (p : X.Walk x y) (q : X.Walk y z) (r : X.Walk z x) : Prop := | |
| ∀ a ∈ p.support, ∃ b, (b ∈ q.support ∨ b ∈ r.support) ∧ X.dist a b ≤ δ | |
| /-- A finitely generated group with a unit-edge Cayley graph whose geodesic triangles | |
| are uniformly thin. Connectedness of this Cayley graph asserts that `S` generates. -/ | |
| def WordHyperbolic (G : Type u) [Group G] : Prop := | |
| ∃ S : Set G, S.Finite ∧ (SimpleGraph.mulCayley S).Connected ∧ | |
| ∃ δ : ℕ, ∀ (x y z : G) | |
| (p : (SimpleGraph.mulCayley S).Walk x y) | |
| (q : (SimpleGraph.mulCayley S).Walk y z) | |
| (r : (SimpleGraph.mulCayley S).Walk z x), | |
| Geodesic (SimpleGraph.mulCayley S) p → | |
| Geodesic (SimpleGraph.mulCayley S) q → | |
| Geodesic (SimpleGraph.mulCayley S) r → | |
| SideThin (SimpleGraph.mulCayley S) δ p q r ∧ | |
| SideThin (SimpleGraph.mulCayley S) δ q r p ∧ | |
| SideThin (SimpleGraph.mulCayley S) δ r p q | |
| end Release075 | |
| end OAI | |