hyperbolic space are real trees. This property characterises hyperbolic spaces. The boundary of a hyperbolic space Generalising the construction of the ends of a simplicial tree there is a natural notion of boundary at infinity for hyperbolic spaces, which has proven very useful for analysing group actions. In this paragraph is a geodesic metric space which is hyperbolic. Definition using the Gromov product A sequence is said to converge to infinity if for some (or any) point we have that as both and go to infinity. Two sequences converging to infinity are considered equivalent when (for some or any ). The boundary of is the set of equivalence classes of sequences which converge to infinity, which is denoted . If are two points on the boundary then their Gromov product is defined to be: which is finite iff . One can then define a topology on using the functions . This topology on is metrisable and there is a distinguished family of metrics defined using the Gromov product. Definition for proper spaces using rays Let be two quasi-isometric embeddings of into ("quasi-geodesic rays"). They are considered equivalent if and only if the function is bounded on . If the space is proper then the set of all such embeddings modulo equivalence with its natural topology is homeomorphic to as defined above. A similar realisation is to fix a basepoint and consider only quasi-geodesic rays originating from this point. In case is geodesic and proper one can also restrict to genuine geodesic rays. Examples When is a simplicial regular tree the boundary is just the space of ends, which is a Cantor set. Fixing a point yields a natural distance on : two points represented by rays originating at are at distance . When is the unit disk, i.e. the Poincaré disk model for the hyperbolic plane, the hyperbolic metric on the disk is and the Gromov boundary can be identified with the unit circle. The boundary of -dimensional hyperbolic space is homeomorphic to the -dimensional sphere and the metrics are similar to the one above. Busemann functions If is proper then its boundary is homeomorphic to the space of Busemann functions on modulo translations. The action of isometries on the boundary and their classification A quasi-isometry between two hyperbolic spaces induces a homeomorphism between the boundaries. In particular the group of isometries of acts by homeomorphisms on . This action