Volume 9, issue 1 (2005)

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Flows and joins of metric spaces

Igor Mineyev

Geometry & Topology 9 (2005) 403–482

arXiv: math.MG/0503274


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 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 X̄ = X X. They are continuous, Isom(X)–invariant, and satisfy sharp identities. We characterize the translation length of a hyperbolic isometry g Isom(X).

For any hyperbolic complex X, the symmetric join X̄ of X̄ and the (generalized) metric d on it are defined. The geodesic flow space (X) arises as a part of X̄. ((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
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
Forward citations
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
Igor Mineyev
Department of Mathematics, University of Illinois at Urbana-Champaign
250 Altgeld Hall
1409 W Green Street
Illinois 61801