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

Abstract

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.