Pseudometric spaces. From minimality to maximality in the groups of combinatorial self-similarities

Abstract

The group of combinatorial self-similarities of a pseudometric space $(X,d)$ is the maximal subgroup of the symmetric group $\mathbf{Sym}(X)$ whose elements preserve the four-point equality $d(x,y)=d(u,v)$. Let us denote by $\mathcal{I}\mathcal{P}$ the class of all pseudometric spaces $(X,d)$ for which every combinatorial self-similarity $\Phi :X\to X$ satisfies the equality $d(x,\Phi (x))=0,$ but all permutations of metric reflection of $(X,d)$ are combinatorial self-similarities of this reflection. The structure of $\mathcal{I}\mathcal{P}$ spaces is fully described.

arXiv comment30 pages