Amenability, Folner sets, and cooling functions

TL;DR

This paper investigates the relationships between Folner ratio, cooling functions, and cooling norms in countable groups, aiming to deepen the theoretical understanding and algorithmic approaches to group amenability.

Contribution

It establishes tight bounds among three complexity measures for finite subsets and explores their algorithmic computation, advancing the theoretical framework for analyzing group amenability.

Findings

Bounded the three measures for finite subsets

Analyzed algorithmic computation of measures

Provided theoretical foundation for group amenability exploration

Abstract

Erling Folner proved that the amenability or nonamenability of a countable group depends on the complexity of its finite subsets. Complexity has three measures: maximum Folner ratio, optimal cooling function, and minimum cooling norm. Our first aim is to show that, for a fixed finite subset, these three measures are tightly bound to one another. We then explore their algorithmic calculation. Our intent is to provide a theoretical background for algorithmically exploring the amenability and nonamenability of discrete groups.