Mathematics – Metric Geometry
Scientific paper
2005-03-14
Geom. Topol. 9(2005) 403-482
Mathematics
Metric Geometry
Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper13.abs.html
Scientific paper
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^n 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-bar= X union bdry 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-bar of X-bar and the (generalized) metric d* on it are defined. The geodesic flow space F(X) arises as a part of *X-bar. (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.
No associations
LandOfFree
Flows and joins of metric spaces does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Flows and joins of metric spaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Flows and joins of metric spaces will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-316372