Flows and joins of metric spaces

We introduce the functor ◦∗ which assigns to every metric space X its symmetric join ◦∗X . As a set, ◦∗X is a union of intervals connecting ordered pairs of points in X . Topologically, ◦∗X is a natural quotient of the usual join of X with itself. We define an Isom(X)–invariant metric d∗ on ◦∗X . Classical concepts known for H 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 ⊔ ∂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 ◦∗X̄ of X̄ and the (generalized) metric d∗ on it are defined. The geodesic flow space F(X) arises as a part of ◦∗X̄ . (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 ◦∗ 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. AMS Classification numbers Primary: 20F65, 20F67, 37D40, 51F99, 57Q05 Secondary: 57M07, 57N16, 57Q91, 05C25