We introduce the functor which assigns to every metric space its symmetric join . As a set, is a union of intervals connecting ordered pairs of points in . Topologically, is a natural quotient of the usual join of with itself. We define an –invariant metric on .
Classical concepts known for and negatively curved manifolds are defined in a precise way for any hyperbolic complex , for example for a Cayley graph of a Gromov hyperbolic group. We define a double difference, a cross-ratio and horofunctions in the compactification . They are continuous, –invariant, and satisfy sharp identities. We characterize the translation length of a hyperbolic isometry .
For any hyperbolic complex , the symmetric join of and the (generalized) metric on it are defined. The geodesic flow space arises as a part of . 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 and has sharp properties. We also give a construction of the asymmetric join of two metric spaces.
These concepts are canonical, ie functorial in , and involve no “quasi"-language. Applications and relation to the Borel conjecture and others are discussed.
"Flows and joins of metric spaces." Geom. Topol. 9 (1) 403 - 482, 2005. https://doi.org/10.2140/gt.2005.9.403