On the density of Cayley graphs of R.Thompson’s group F in symmetric generators

By the density of a finite graph we mean its average vertex degree. For an [Formula: see text]-generated group, Cayley in given set generators, is supremum densities taken over all subgraphs. It known that group with text] generators amenable if and only corresponding equals text]. A famous problem on amenability R. Thompson’s still open. Due to result Belk Brown, it standard text], at least This estimate has not been exceeded so far. symmetric where same example gave There was conjecture for this generating equality holds. If so, would be non-amenable, have doubling property. any inequality In paper, disprove showing strictly exceeds Moreover, show even larger does