Buckley, Stephen M. and Kokkendorff, Simon L.
(2008)
Detours and Gromov hyperbolicity.
International Journal of Pure and Applied Mathematics  IJPAM, 47.
pp. 313323.
ISSN 13118080
Abstract
The notion of Gromov hyperbolicity was introduced by Gromov in the setting of geometric group
theory [G1], [G2], but has played an increasing role in analysis on general metric spaces [BHK],
[BS], [BBo], [BBu], and extendability of Lipschitz mappings [L].
In this theory, it is often additionally assumed that the hyperbolic metric space is proper and
geodesic (meaning that closed balls are compact, and each pair of points can be joined by a path
whose length equals the distance between the points). These additional assumptions are useful
in proofs, and valid for large classes of examples of hyperbolic spaces, for instance Cayley graphs
of (nitely generated) hyperbolic groups, and certain important conformal distortions of locally
compact length metrics that push the boundary of the space to innity; see for instance [BHK, 2.8]
for the case of a quasihyperbolic metric. However if the underlying metric is not locally compact,
as in examples that arise in a Banach space context, then such hyperbolic conformal distortions
typically fail to be proper and geodesic (although they are always length spaces).
Without these added assumptions, a few \standard" results for hyperbolic spaces may fail; for
one such example, see [GH, 5.13]. However, Vaisala recently proved [V1] that a large part of the
theory of hyperbolic spaces goes through if we merely assume the metric is a length metric and not
geodesic or proper; Vaisala then applied this theory in a Banach space context [V2].
Our paper adds to the work of [V1] by extending a characterization by Bonk of hyperbolicity
in a geodesic context [B] to a length space context; see Theorem 2.1 below. Note that our version
says a little more than Bonk's result even in a geodesic space context.
Item Type: 
Article

Keywords: 
Detours; Gromov; hyperbolicity; 
Academic Unit: 
Faculty of Science and Engineering > Mathematics and Statistics 
Item ID: 
1573 
Depositing User: 
Prof. Stephen Buckley

Date Deposited: 
07 Oct 2009 15:41 
Journal or Publication Title: 
International Journal of Pure and Applied Mathematics  IJPAM 
Refereed: 
Yes 
URI: 

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