On the Ore condition for the group ring of R.\,Thompson's group $F$

Abstract

Let $R=K[G]$ be a group ring of a group $G$ over a field $K$. The Ore condition says that for any $a,b\in R$ there exist $u,v\in R$ such that $au=bv$, where $u\ne0$ or $v\ne0$. It always holds whenever $G$ is amenable. Recently it was shown that for R.\,Thompson's group $F$ the converse is also true. So the famous amenability problem for $F$ is equivalent to the question on the Ore condition for the group ring of the same group. It is easy to see that the problem on the Ore condition for $K[F]$ is equivalent to the same property for the monoid ring $K[M]$, where $M$ is the monoid of positive elements of $F$. In this paper we reduce the problem to the case when $a$, $b$ are homogeneous elements of the same degree in the monoid ring. We study the case of degree $1$ and find solutions of the Ore equation. For the case of degree $2$, we study the case of linear combinations of monomials from $S=\{x_0^2,x_0x_1,x_0x_2,x_1^2,x_1x_2\}$. This set is not doubling, that is, there are nonempty finite subsets $X\subset M\subset F$ such that $|SX| < 2|X|$. As a consequence, the Ore condition holds for linear combinations of these monomials. We give an estimate for the degree of $u$, $v$ in the above equation. The case of monomials of higher degree is open as well as the case of degree $2$ for monomials on $x_0,x_1,...,x_m$, where $m\ge3$. Recall that negative answer to any of these questions will immediately imply non-amenability of $F$.