Geometry & Topology 9 (2005) 403–482

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 Hⁿ 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=X⊔∂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 of X and the (generalized) metric d_* on it are defined. The geodesic flow space F(X) arises as a part of *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

Primary: 20F65, 20F67, 37D40, 51F99, 57Q05

Secondary: 05C25, 57M07, 57N16, 57Q91

References

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