Proper Gromov transforms of metrics are metrics. (English) Zbl 1040.54016
In phylogenetic analysis, a standard problem is to approximate a given metric (usually derived from some data set) by a tree metric. Such a method was proposed by Gromov in the context of hyperbolic groups in [M. Gromov, Hyperbolic Groups, Essays in group theory, Publ., Math. Sci. Res. Inst. 8, 75–263 (1987; Zbl 0634.20015)], leading to the Gromov transform which is considered in this paper. In particular, the main result of this paper is to characterize those maps whose Gromov transforms are metrics. It is expected that this result will help to better understand some tree reconstruction procedures used in phylogenetic analysis.
Reviewer: Vincent Moulton (Uppsala)
Keywords:
metrics; additive metrics; ultra metrics; Farris transform; phylogenetic analysis; phylogenetic combinatoricsCitations:
Zbl 0634.20015References:
| [1] | A. Dress, K.T. Huber, J.H. Koolen, V. Moulton and J. Weyer-Menkhoff, Δ additive and Δ ultra-additive maps, Gromov’s trees, and the Farris transform (submitted).; A. Dress, K.T. Huber, J.H. Koolen, V. Moulton and J. Weyer-Menkhoff, Δ additive and Δ ultra-additive maps, Gromov’s trees, and the Farris transform (submitted). · Zbl 1056.92042 |
| [2] | Gromov, M., Hyperbolic Groups, (Essays in Group Theory, MSRI Series, Volume 8 (1988), Springer-Verlag) · Zbl 0634.20015 |
| [3] | Bowditch, B., Notes on Gromov’s hyperbolicity criterion for path metric spaces, (Ghys, E., Group Theory From a Geometric Viewpoint (1991), World Scientific), 64-167 · Zbl 0843.20031 |
| [4] | Farris, J. S., On the phenetic approach to vertebrate classification, (Major Patterns in Vertebrate Evolution (1977), Plenum: Plenum New York), 823-950 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
