Spaces that can be ordered effectively: virtually free groups and hyperbolicity

TL;DR

The paper investigates how metric spaces and groups can be ordered efficiently based on asymptotic invariants related to the traveling salesman problem, highlighting characterizations of virtually free groups and hyperbolic spaces.

Contribution

It characterizes virtually free groups via efficient ordering of 4-point subsets and demonstrates that hyperbolic spaces can be ordered very efficiently as the subset size grows.

Findings

Virtually free groups admit efficient orderings on 4-point subsets.

Hyperbolic spaces can be ordered extremely efficiently for large subsets.

Provides a classification of groups and spaces based on ordering efficiency.

Abstract

We study asymptotic invariants of metric spaces, defined in terms of the travelling salesman problem, and our goal is to classify groups and spaces depending on how well they can be ordered in this context. We characterize virtually free groups as those admitting an order which has some efficiency on $4$-point subsets. We show that all $\delta$-hyperbolic spaces can be ordered extremely efficiently, for the question when the number of points of a subset tends to $\infty$.