#### Volume 9, issue 1 (2005)

 Download this article For screen For printing
 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1364-0380 ISSN (print): 1465-3060
Flows and joins of metric spaces

### Igor Mineyev

Geometry & Topology 9 (2005) 403–482
 arXiv: math.MG/0503274
##### Abstract

We introduce the functor $\ast$ which assigns to every metric space $X$ its symmetric join $\ast X$. As a set, $\ast X$ is a union of intervals connecting ordered pairs of points in $X$. Topologically, $\ast X$ is a natural quotient of the usual join of $X$ with itself. We define an $Isom\left(X\right)$–invariant metric ${d}_{\ast }$ on $\ast X$.

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 $\stackrel{̄}{X}=X\bigsqcup \partial X$. They are continuous, $Isom\left(X\right)$–invariant, and satisfy sharp identities. We characterize the translation length of a hyperbolic isometry $g\in Isom\left(X\right)$.

For any hyperbolic complex $X$, the symmetric join $\ast \stackrel{̄}{X}$ of $\stackrel{̄}{X}$ and the (generalized) metric ${d}_{\ast }$ on it are defined. The geodesic flow space $\mathsc{ℱ}\left(X\right)$ arises as a part of $\ast \stackrel{̄}{X}$. $\left(\mathsc{ℱ}\left(X\right),{d}_{\ast }\right)$ 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 $X\ast Y$ 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.

##### Keywords
symmetric join, asymmetric join, metric join, Gromov hyperbolic space, hyperbolic complex, geodesic flow, translation length, geodesic, metric geometry, double difference, cross-ratio
##### Mathematical Subject Classification 2000
Primary: 20F65, 20F67, 37D40, 51F99, 57Q05
Secondary: 57M07, 57N16, 57Q91, 05C25
##### Publication
Received: 29 July 2004
Revised: 17 February 2005
Accepted: 22 February 2005
Published: 9 March 2005
Corrected: 9 January 2009 (link on page 479 updated)
Proposed: Walter Neumann
Seconded: Martin Bridson, David Gabai
##### Authors
 Igor Mineyev Department of Mathematics, University of Illinois at Urbana-Champaign 250 Altgeld Hall 1409 W Green Street Urbana Illinois 61801 USA