How rigid the finite ultrametric spaces can be?

TL;DR

This paper investigates the rigidity of finite ultrametric spaces, characterizing those with minimal isometry groups and exploring their extremal properties and graph-theoretic representations.

Contribution

It provides a characterization of finite ultrametric spaces where every self-isometry fixes all but at most two points, using representing trees and graph theory.

Findings

Finite ultrametric spaces with minimal isometry groups are characterized.

Representing trees are used to analyze the isometry properties.

Extremal properties and graph characterizations of these spaces are established.

Abstract

A metric space $X$ is rigid if the isometry group of $X$ is trivial. The finite ultrametric spaces $X$ with $|X| \geq 2$ are not rigid since for every such $X$ there is a self-isometry having exactly $|X|-2$ fixed points. Using the representing trees we characterize the finite ultrametric spaces $X$ for which every self-isometry has at least $|X|-2$ fixed points. Some other extremal properties of such spaces and related graph theoretical characterizations are also obtained.