Flows and joins of metric spaces

Mineyev, Igor

doi:10.2140/gt.2005.9.403

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Flows and joins of metric spaces

Igor Mineyev

Geometry & Topology 9 (2005) 403–482

DOI: 10.2140/gt.2005.9.403

arXiv: math.MG/0503274

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 $\bar{X} = X ⊔ \partial X$ . 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 $* \bar{X}$ of $\bar{X}$ and the (generalized) metric $d_{*}$ on it are defined. The geodesic flow space $ℱ (X)$ arises as a part of $* \bar{X}$ . $(ℱ (X), d_{*})$ 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.

Additional material

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

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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

Volume 9, issue 1 (2005)