a Gromov-hyperbolic space
/ GROH-mof /
If you zoom far enough out from the hyperbolic plane, or from a tree, you notice that triangles are 'thin' — any one side hugs close to the other two, and there is no room for a big flat region. Gromov took this large-scale thinness and made it the entire definition. A Gromov-hyperbolic space is a metric space whose geodesic triangles are uniformly thin, capturing negative curvature in a way that survives the coarse, distort-but-don't-tear viewpoint of geometric group theory.
The cleanest version is the thin-triangles condition. A geodesic space X is delta-hyperbolic (for some constant delta >= 0) if every geodesic triangle is delta-thin: each side lies within the delta-neighborhood of the union of the other two. So if you stand anywhere on one side of a triangle, you are within distance delta of some point on another side — the triangle has no fat middle. Equivalently one can use the four-point condition on the Gromov product (x.y)_w = (1/2)(d(x,w) + d(y,w) - d(x,y)): for all points, (x.z)_w >= min{(x.y)_w, (y.z)_w} - delta. A tree is 0-hyperbolic; the hyperbolic plane is delta-hyperbolic for a specific delta; the value of delta does not matter, only that some finite delta works. Crucially the property is a quasi-isometry invariant: a space quasi-isometric to a hyperbolic space is itself hyperbolic, which is why it is the right notion for groups.
Gromov-hyperbolicity is the large-scale backbone of negative curvature and the foundation of hyperbolic group theory. A hyperbolic space has a well-defined boundary at infinity carrying a visual metric, geodesics between nearby points stay uniformly close (the Morse lemma: quasi-geodesics track real geodesics), and the geometry forces a solvable word problem when the space is a group's Cayley graph. The decisive caveat separating it from CAT(0): hyperbolicity is coarse and allows bounded flat pieces, but forbids arbitrarily large flats. Euclidean space R^n for n >= 2 is NOT hyperbolic — its triangles get arbitrarily fat as they grow — even though it is CAT(0). Conversely a hyperbolic space need not be CAT(0) or even geodesically nice locally; it only constrains large-scale shape.
A regular tree of degree 3 (every vertex meets three edges) is 0-hyperbolic: any geodesic triangle is a tripod, so each side actually lies inside the union of the other two, delta = 0. The free group on two generators has exactly this tree as its Cayley graph, so it is a hyperbolic group. By contrast, the integer grid Z^2 is not hyperbolic: a large square has a corner point on one side at distance roughly half the side length from the other two sides, and that distance grows without bound, so no single delta works.
A tree is 0-hyperbolic (triangles are tripods); the flat grid Z^2 is not hyperbolic.
Hyperbolicity is a coarse, large-scale property — it says nothing about local structure and is invariant under quasi-isometry. The specific value of delta is not meaningful; only its existence is. And R^n (n >= 2) is CAT(0) yet not hyperbolic, so the two curvature notions are independent.