Computable Folner sequences of amenable groups
Karol Duda, Aleksander Ivanov
TL;DR
This work develops a comprehensive account of computable amenability, extending Cavaleri’s results from finitely generated groups to computably enumerable groups without finite generation and establishing equivalences among amenability, computable Reiter functions, subrecursive Følner functions, and $\Sigma$-amenability. It further analyzes the algorithmic complexity of effective Følner sequences, showing the set of computable Følner sequences is in $\Pi^0_3$ and that this problem is $\Pi^0_3$-complete in certain abelian cases, as well as examining the nontrivial convergence moduli of the associated means. The paper also introduces a framework for computable presentations of metric groups, proving that amenability and computable amenability coincide in the computable metric group setting and extending the discrete results to computable metric groups via Schneider–Thom style criteria. Together, these results provide a robust, algorithmic perspective on amenability applicable to both discrete and metric groups, with implications for invariant means and the structure of Folner sequences in computable settings.
Abstract
The paper considers computable Folner sequences in computably enumerable amenable groups. We extend some basic results of M. Cavaleri on existence of such sequences to the case of groups where finite generation is not assumed. We also initiate some new directions in this topic, for example complexity of families of effective Folner sequences. Possible extensions of this approach to metric groups are also discussed. This paper also contains some unpublished results from the paper of the first author arXiv:1904.02640.