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2005 Flows and joins of metric spaces
Igor Mineyev
Geom. Topol. 9(1): 403-482 (2005). DOI: 10.2140/gt.2005.9.403

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 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=XX. 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 XY 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.

Citation

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Igor Mineyev. "Flows and joins of metric spaces." Geom. Topol. 9 (1) 403 - 482, 2005. https://doi.org/10.2140/gt.2005.9.403

Information

Received: 29 July 2004; Revised: 17 February 2005; Accepted: 22 February 2005; Published: 2005
First available in Project Euclid: 20 December 2017

zbMATH: 1137.37314
MathSciNet: MR2140987
Digital Object Identifier: 10.2140/gt.2005.9.403

Subjects:
Primary: 20F65 , 20F67 , 37D40 , 51F99 , 57Q05
Secondary: 05C25 , 57M07 , 57N16 , 57Q91

Keywords: asymmetric join , cross-ratio , double difference , Geodesic , geodesic flow , Gromov hyperbolic space , hyperbolic complex , metric geometry , metric join , symmetric join , translation length

Rights: Copyright © 2005 Mathematical Sciences Publishers

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