Gromov product structures, quadrangle structures and split metric spaces

Abstract

Let (X,d) be a finite metric space with elements Pi, i=1,…,n and with distances dij≔d(Pi,Pj) for i,j=1,…,n. The “Gromov product” Δijk, is defined as [Formula presented]. (X,d) is called Δ-generic, if, for each fixed i, the set of Gromov products Δijk has a unique smallest element, Δijiki. The Gromov product structure on a Δ-generic finite metric space (X,d) is the map that assigns the edge Ejiki to Pi. A finite metric space is called “quadrangle generic”, if for all 4-point subsets {Pi,Pj,Pk,Pl}, the set {dij+dkl,dik+djl,dil+djk} has a unique maximal element. The “quadrangle structure” on a quadrangle generic finite metric space (X,d) is defined as a map that assigns to each 4-point subset of X the pair of edges corresponding to the maximal element of the sums of distances. Two metric spaces (X,d) and (X,d′) are said to be Δ-equivalent (Q-equivalent), if the corresponding Gromov product (quadrangle) structures are the same up to a permutation of X. We show that Gromov product classification is coarser than the metric fan classification. Furthermore it is proved that: (i) The isolation index of the 1-split metric δi is equal to the minimal Gromov product at the vertex Pi. (ii) For a quadrangle generic (X,d), the isolation index of the 2-split metric δij is nonzero if and only if the edge Eij is a side in every quadrangle whose set of vertices includes Pi and Pj. (iii) For a quadrangle generic (X,d), the isolation index of an m-split metric δi1…im is nonzero if and only if any edge Eikil is a side in every quadrangle whose vertex set contains Pik and Pil. These results are applied to construct a totally split decomposable metric for n=6.