# F{\o}lner functions and the generic Word Problem for finitely generated amenable groups

## TL;DR

This paper explores effective amenability in finitely generated groups through F{\

## Contribution

It introduces new definitions of effective amenability and demonstrates the existence of finitely presented groups with unsolvable generic Word Problem.

## Key findings

- Recursively presented amenable groups have subrecursive F{\
- Solvability of generic Word Problem does not imply solvability of generic Equality Problem in finitely presented groups.
- First examples of finitely presented groups with unsolvable generic Equality Problem.

## Abstract

We introduce and investigate different definitions of effective amenability, in terms of computability of F{\o}lner sets, Reiter functions, and F{\o}lner functions. As a consequence, we prove that recursively presented amenable groups have subrecursive F{\o}lner function, answering a question of Gromov, for the same class of groups we prove that solvability of the Equality Problem on a generic set (generic EP) is equivalent to solvability of the Word Problem on the whole group (WP), thus providing the first examples of finitely presented groups with unsolvable generic EP. In particular, we prove that for finitely presented groups, solvability of generic WP doesn't imply solvability of generic EP.