# Superharmonic functions on the Lamplighter graph of Thompson's group F

## TL;DR

This paper explores a new criterion for group amenability using superharmonic functions and random walks, applied specifically to Thompson's group F, aiming to advance understanding of its amenability status.

## Contribution

It introduces a novel approach extending amenability tests to group actions and applies it to Thompson's group F, providing a new perspective and tools for this longstanding problem.

## Key findings

- Developed a new criterion for group amenability based on superharmonic functions.
- Applied the criterion to Thompson's group F, offering insights into its properties.
- Tested the approach on a subclass of superharmonic functions, advancing the methodology.

## Abstract

The goal is to extend a non-standard amenability test for groups, based on random walks and superharmonic functions, to group actions on sets, and to apply it to Thompson's group F using certain properties of extensive amenability. While no conclusive answer regarding the amenability of F is given, the approach is helpful in developing a new potentially useful criterion and testing it on a significant subclass of superharmonic functions.