Amenability of Thompson's group F

Is Thompson's group \(F\) — the group of piecewise-linear dyadic homeomorphisms of the unit interval — amenable as a discrete group?

Thompson introduced the groups F, T, V in the 1960s (in logic); Higman–Brown developed them geometrically. Geoghegan conjectured F is non-amenable. Evidence cuts both ways: Brin–Squier (1985) showed F contains no non-abelian free subgroup (so von Neumann's conjecture, were it true, would settle it — but Olshanskii–Olshanskii–Sapir refuted that route), while F is not elementary amenable and has exponential growth. Monod (2013) built the first non-amenable groups without free subgroups, proving the missing-free-subgroup evidence inconclusive. Amenability of F itself — where an invariant mean would have to exploit self-similarity rather than subexponential growth — remains open.