Towards a classification of transitive group actions on finite metric spaces
type: Item
instance of: scholarly article
title: Towards a classification of transitive group actions on finite metric spaces
zbMATH Open document ID: 0683.54041
DOI: 10.1016/0001-8708(89)90008-X
published in: Advances in Mathematics
publication date: 1989-01-01T00:00:00Z
review text: The author investigates finite metric spaces \(X\) together with transitive groups G of isometries of \(X\). The main tool in the injective hull \(T_ X\) of \(X\)constructed by \textit{J. R. Isbell} [Comment. Math. Helv. 39, 65-74 (1964; Zbl 0149.015)], which can be viewed as a certain compact polyhedron in \({\mathbb{R}}^ X\). It is shown that the dimension of \(T_ X\) is at most \([{\#}X/2]\). In what follows, assume throughout that equality holds. Then the number of facets of maximal dimension \(n\) in \(T_ X\) is 1 if \({\#}X=2n\), and is at most if \({\#} X=2n+1\). These are the basic ingredients for the rather involved proof of Theorem 1: Let \(G\) act transitively on a set \(X\). Then a \(G\)-invariant metric on \(X\) satisfying the above assumptions exist if and only if either \({\#}X\) is even and there is a fixed point free involution \(\sigma\) on \(X\) centralizing the action of \(G\), or \({\#}X\) is odd and \(G\) is cyclic or dihedral. So by the Feit-Thompson theorem, every finite simple group admits a metric of this kind invariant under left multiplication, \(\sigma\) being right multiplication by an involution in \(C\). In the even case, the involution is obtained from the unique maximal dimensional facet of \(T_ X\), that has to be \(G\)-invariant. The complicated part of the proof concerns the odd case. The author calls a bijection between metric spaces a quasi-isometry if it induces a bijection of the sets of maximal dimension facets of their injective hulls. Clearly an isometry is an quasi-isometry. An essential step in the proof of Theorem 1 is Theorem 2 which associates to every odd \(X\) a graph \(E\) with vertex set \(X\) and \({\#}X\) edges, and an involution \(\tau\) of \(E\), such that the quasi-isometries of \(X\) onto \(X'\) are precisely the bijectionsmapping \((E,\tau)\) onto \((E',\tau').\) The reviewer thinks that in the even case, Theorem 1 indicates that classification is hopeless, since any finite transitive action can be ``double…
Mathematics Subject Classification ID: 54H15
Mathematics Subject Classification ID: 54E40
Mathematics Subject Classification ID: 20B25
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 52A37
Mathematics Subject Classification ID: 52Bxx
zbMATH DE Number: 4119226
zbMATH Keywords: transitive groups of isometries
zbMATH Keywords: finite metric spaces
zbMATH Keywords: injective hull
zbMATH Keywords: G- invariant metric
zbMATH Keywords: invariant under left multiplication
zbMATH Keywords: quasi-isometry
zbMATH Keywords: sets of maximal dimension facets
author: Andreas W. M. Dress
MaRDI profile type: Publication
cites work: Abstract length functions in groups
cites work: Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: A note on combinatorial properties of metric spaces
cites work: Six theorems about injective metric spaces
cites work: Length Functions in Groups.
description: scientific article; zbMATH DE number 4119226