Gromov hyperbolicity: A group G is hyperbolic (in the sense of Gromov) if its Cayley graph satisfies the δ-slim triangles condition — all geodesic triangles have each side within δ of the union of the other two.
Free groups F_n are δ-hyperbolic with δ=0 (tree-like). The Gromov boundary ∂G is the space of geodesic rays, homeomorphic to a Cantor set for free groups.
In the Poincaré disk model, the hyperbolic plane has curvature K<0; edges between group elements appear as hyperbolic geodesics (arcs of circles orthogonal to the boundary).
The exponential growth of the ball of radius r — |B(r)| ~ e^{h·r} where h is the volume entropy — is characteristic of hyperbolic groups.