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544609e | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 | 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
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