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This is a survey of our recent results on the amenability problem forThompson's group $F$. They mostly concern esimating the density of finitesubgraphs in Cayley graphs of $F$ for various systems of generators, and alsoequations in the group ring of $F$. We also discuss possible approaches tosolve the problem in both directions.

Amenability problem for Thompson's group $F$: state of the art