the boundary at infinity
Stand inside a negatively curved world and look out in every direction toward the horizon. Each direction you could walk off to forever is a point 'at infinity,' and the set of all such directions wraps the space in an invisible sphere you can never reach but can always see. The boundary at infinity is exactly this set of endpoints of geodesics going to infinity, attached to a hyperbolic or nonpositively curved space as a kind of horizon. It is where the large-scale geometry crystallizes into a compact object you can study directly.
Concretely, fix a Gromov-hyperbolic space X and a basepoint w. A geodesic ray from w runs off to infinity; declare two rays equivalent if they stay a bounded distance apart forever. The boundary at infinity, written ∂X or ∂_infinity X, is the set of these equivalence classes. There is a cleaner equivalent built from the Gromov product: a sequence x_n 'converges to infinity' if (x_m . x_n)_w -> infinity as m, n -> infinity, and two such sequences define the same boundary point when their Gromov product also tends to infinity. The boundary carries a natural family of visual metrics: roughly, two boundary points are close when the geodesics toward them stay together for a long time before diverging, formalized as a distance like exp(-epsilon (xi . eta)_w) where (xi . eta)_w measures how long the rays fellow-travel. These metrics depend on the basepoint and a parameter epsilon, but only up to a quasi-symmetric change.
The boundary at infinity converts coarse geometry into a tractable topological-and-metric space, and it is a quasi-isometry invariant up to the right equivalence: a quasi-isometry between hyperbolic spaces extends to a quasi-symmetric homeomorphism of their boundaries. This is the engine of rigidity in geometric group theory — the boundary of a hyperbolic group is a well-defined topological space, and its homeomorphism type constrains the group. For example the boundary of the hyperbolic plane is a circle, and of hyperbolic n-space an (n-1)-sphere; free groups have a Cantor set boundary. A caution: the visual metric is canonical only up to quasi-symmetry, not isometry, so one studies the boundary as a quasi-symmetry-class of metric spaces, and the metric structure depends on choices the topology does not.
For the hyperbolic plane modeled as the open unit disc (Poincare disc), the boundary at infinity is literally the bounding circle: every geodesic ray heads toward a unique point on the circle, and two rays converge to the same boundary point exactly when they aim at the same circle point. The visual metric on this circle is comparable to ordinary arc length. For the free group F_2, whose Cayley graph is the 4-valent tree, the boundary is a Cantor set — the ends of the tree.
Boundary of the hyperbolic plane is a circle; boundary of the free group's tree is a Cantor set.
The boundary is canonical as a topological space, but its visual metric is only canonical up to quasi-symmetry — there is no single 'the' visual metric, since it depends on a basepoint and a parameter. Conflating the topological boundary with a fixed metric is a common slip.