Characterizing Hyperbolic Spaces and Real Trees

Let X be a geodesic metric space. Gromov proved that there exists ε0 > 0 such that if every sufficiently large triangle ∆ satisfies the Rips condition with constant ε0 · pr(∆), where pr(∆) is the perimeter ∆, then X is hyperbolic. We give an elementary proof of this fact, also giving an estimate for ε0. We also show that if all the triangles ∆ ⊆ X satisfy the Rips condition with constant ε0 · pr(∆), then X is a real tree. Moreover, we point out how this characterization of hyperbolicity can be used to improve a result by Bonk, and to provide an easy proof of the (well-known) fact that X is hyperbolic if and only if every asymptotic cone of X is a real tree. 1. Preliminaries and statements Let (X, d) be a metric space. A map γ : [0, 1] → X is a geodesic if there exists k ≥ 0 such that d(γ(t), γ(s)) = k|t−s| for every t, s ∈ [0, 1]. The space X is geodesic if any pair of points in X can be connected by a geodesic, and uniquely geodesic if such a geodesic is unique. With an abuse, we identify geodesics and their images, and we let [x, y] denote a geodesic joining x to y, even though this geodesic is not unique. A triangle with vertices x, y, z is the union of three geodesics [x, y], [y, z], [z, x], called sides, and will be denoted by ∆(x, y, z). We denote by pr(∆) the perimeter of ∆, i.e. we set pr(∆(x, y, z)) = d(x, y) + d(y, z) + d(z, x). 1.1. Gromov hyperbolic spaces and real trees. For A ⊆ X and ε > 0, we set Nε(A) = {x ∈ X : d(x,A) ≤ ε} A triangle with sides l1, l2, l3 satisfies the Rips condition with constant δ if for {i, j, k} = {1, 2, 3} we have li ⊆ Nδ(lj ∪ lk). A geodesic space X is δ-hyperbolic if every triangle in X satisfies the Rips condition with constant δ, and it is hyperbolic if it is δ-hyperbolic for some δ ≥ 0. A 0-hyperbolic geodesic space is also called a real tree. It is easily seen that a real tree is uniquely geodesic, and that if [x, y], [y, x] are geodesics in a real tree such that [x, y] ∩ [y, z] = {y}, then [x, z] = [x, y] ∪ [y, z]. 1.2. The main results. Let X be a fixed geodesic space. For every triangle ∆ in X we provide a measure of how much non-hyperbolic ∆ is by setting δ(∆) = inf{δ : ∆ satisfies the Rips condition with constant δ}. 2000 Mathematics Subject Classification. 53C23, 20F67 (secondary).