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.

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Igor Mineyev
Department of Mathematics, University of Illinois at Urbana-Champaign
250 Altgeld Hall, 1409 W Green Street, Urbana, IL 61801, USA

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