This is a survey of our recent results on the amenability problem for Thompson's group $F$. They mostly concern esimating the density of finite subgraphs in Cayley graphs of $F$ for various systems of generators, and also equations in the group ring of $F$. We also discuss possible approaches to solve the problem in both directions.
Comment: 24 pages, 1 figure. Published in journal of Groups, Complexity, Cryptology. arXiv admin note: text overlap with arXiv:2112.09812, arXiv:2210.12304