Flows and joins of metric spaces

We introduce the functor circled asterisk operator sign which assigns to every metric space X its symmetric join circled asterisk operator sign X. As a set, circled asterisk operator sign X is a union of intervals connecting ordered pairs of points in X. Topologically, circled asterisk operator sign X is a natural quotient of the usual join of X with itself. We define an Isom(X)-invariant metric d_* on circled asterisk operator sign X. Classical concepts known for ℍⁿ and negatively curved manifolds are defined in a precise way for any hyperbolic complex X, for example for a Cayley graph of a Gromov hyperbolic group. We define a double difference, a cross-ratio and horofunctions in the compactification X̄ = X square union sign ∂ X. They are continuous, Isom(X)- invariant, and satisfy sharp identities. We characterize the translation length of a hyperbolic isometry g ∈ Isom(X). For any hyperbolic complex X, the symmetric join circled asterisk operator sign X̄ of X̄ and the (generalized) metric d _* on it are defined. The geodesic now space F(X) arises as a part of circled asterisk operator sign X. (F(X), d_*) is an analogue of (the total space of) the unit tangent bundle on a simply connected negatively curved manifold. This flow space is defined for any hyperbolic complex X and has sharp properties. We also give a construction of the asymmetric join X circled asterisk operator sign Y of two metric spaces. These concepts are canonical, i.e. functorial in X, and involve no "quasi"-language. Applications and relation to the Borel conjecture and others are discussed.