#### Geometry & Topology, Vol. 9 (2005)
Paper no. 13, pages 403--482.

## Flows and joins of metric spaces

### Igor Mineyev

**Abstract**.
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.
**Keywords**.
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.

**DOI:** 10.2140/gt.2005.9.403

**E-print:** `arXiv:math.MG/0503274`

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
For the LaTeX codes used for the symbols in the paper see:

Igor Mineyev

Department of Mathematics, University of Illinois at Urbana-Champaign

250 Altgeld Hall, 1409 W Green Street, Urbana, IL 61801, USA

Email: mineyev@math.uiuc.edu

GT home page

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