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Towards a classification of transitive group actions on finite metric spaces. (English) Zbl 0683.54041

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 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 “doubled” \((G\times 4Z_ 2\) on \(X\times 3(0,1\})\) so that a transitive action centralized by a free involution results.
Reviewer: R.Löwen

MSC:

54H15 Transformation groups and semigroups (topological aspects)
54E40 Special maps on metric spaces
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
20F65 Geometric group theory
52A37 Other problems of combinatorial convexity
52Bxx Polytopes and polyhedra

Citations:

Zbl 0149.015
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References:

[1] Chiswell, I. M., Abstract length functions in groups, (Math. Proc. Cambridge Philos. Soc., 80 (1976)), 451-463 · Zbl 0351.20024
[2] Dress, A. W.M, Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups, a note on combinatorial properties of metric spaces, Adv. in Math., 53, 3, 321-402 (1984) · Zbl 0562.54041
[3] Isbell, J. R., Six theorems about injective metric spaces, Comment. Math. Helv., 39, 65-74 (1964-1965) · Zbl 0151.30205
[4] Lyndon, R. C., Length functions in groups, Math. Scand., 12, 209-234 (1963) · Zbl 0119.26402
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