Amenability, Folner sets, and cooling functions

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.