Cayley graphs of R. Thompson’s group $F$: New estimates for the density

Abstract

By the density of a finite graph we mean its average vertex degree. For the Cayley graph of a group $G$ with $m$ generators, it is known that $G$ is amenable if and only if the supremum of densities of its finite subgraphs has value $2m$.
For R. Thompson’s group $F$, the problem of its amenability is a long-standing open question. There were several attempts to solve it in both directions. For the Cayley graph of $F$ in the standard set of group generators $\{x_0,x_1\}$, there exists a construction due to Jim Belk and Ken Brown. It was presented in 2004. This is a family of finite subgraphs whose densities approach 3.5. Many unsuccessful attempts to improve this estimate led to the conjecture that this construction was optimal. This would imply non-menability of $F$.
Recently, we have found an improvement showing that this conjecture is false. Namely, there exist finite subgraphs in the Cayley graph of $F$ in generators $x_0$, $x_1$ with density strictly exceeding 3.5. This makes amenability of $F$ more plausible. Besides, we disprove one more conjecture, by showing that there are finite subgraphs in the Cayley graph of $F$ in generators $x_0$, $x_1$, $x_2$ with density strictly exceeding 5.