  By the density of a finite graph we mean its average vertex degree. For an
$m$-generated group, the density of its Cayley graph in a given set of
generators, is the supremum of densities taken over all its finite subgraphs.
It is known that a group with $m$ generators is amenable iff the density of the
corresponding Cayley graph equals $2m$. A famous problem on the amenability of
R.\,Thompson's group $F$ is still open. What is known due to the result by Belk
and Brown, is that the density of its Cayley graph in the standard set of group
generators $\{x_0,x_1\}$, is at least $3.5$. This estimate has not been
exceeded so far. For the set of symmetric generators $S=\{x_1,\bar{x}_1\}$,
where $\bar{x}_1=x_1x_0^{-1}$, the same example gave the estimate only $3$.
There was a conjecture that for this generating set the equality holds. If so,
$F$ would be non-amenable, and the symmetric generating set had doubling
property. This means that for any finite set $X\subset F$, the inequality
$|S^{\pm1}X|\ge2|X|$ holds. In this paper we disprove this conjecture showing
that the density of the Cayley graph of $F$ in symmetric generators $S$
strictly exceeds $3$. Moreover, we show that even larger generating set
$S_0=\{x_0,x_1,\bar{x}_1\}$ does not have doubling property.

Group Theory