On Følner sets in topological groups

We extend Følner’s amenability criterion to the realm of general topological groups. Building on this, we show that a topological group $G$ is amenable if and only if its left-translation action can be approximated in a uniform manner by amenable actions on the set $G$. As applications we obtain a topological version of Whyte’s geometric solution to the von Neumann problem and give an affirmative answer to a question posed by Rosendal.

Computability of Følner sets

We define the notion of computability of Følner sets for finitely generated amenable groups. We prove, by an explicit description, that the Kharlampovich groups, finitely presented solvable groups with unsolvable Word Problem, have computable Følner sets. We also prove computability of Følner sets for extensions — with subrecursive distortion functions — of amenable groups with solvable Word Problem by finitely generated groups with computable Følner sets. Moreover, we obtain some known and some new upper bounds for the Følner function for these particular extensions.

Følner Independence and the amenable Ising model

We define a criterion called Følner Independence for a stationary process over an amenable group. Intuitively, a process satisfies the criterion if, for sufficiently invariant Følner sets, the process in the Følner set is nearly independent of the process outside. We show that Følner Independence implies Finitely Determined. As an application, we show that, in its extreme Gibbs states, the amenable attractive Ising model is Følner Independent (hence Finitely Determined, hence Bernoulli).