Geometry & Topology, Vol. 9 (2005)
Paper no. 13, pages 403--482.
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^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.
Symmetric join, asymmetric join, metric join, Gromov hyperbolic space,
hyperbolic complex, geodesic flow, translation length, geodesic,
metric geometry, double difference, cross-ratio
AMS subject classification.
Primary: 20F65, 20F67, 37D40, 51F99, 57Q05.
Secondary: 57M07, 57N16, 57Q91, 05C25.
Submitted to GT on 29 July 2004.
(Revised 17 February 2005.)
Paper accepted 22 February 2005.
Paper published 9 March 2005.
Notes on file formats
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Department of Mathematics, University of Illinois at Urbana-Champaign
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