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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 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
X=X⊔∂X. They are
continuous, Isom(X)–invariant, and satisfy sharp identities. We
characterize the translation length of a hyperbolic isometry
g in Isom(X).
For any hyperbolic complex X, the symmetric join
*X
of X and the
(generalized) metric d* on it are defined. The
geodesic flow space F(X) arises as a part of *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*Y of two metric
spaces.
These concepts are canonical, ie functorial in X, and involve no
"quasi"-language. Applications and relation to the Borel conjecture
and others are discussed.
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