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

TL;DR

This paper investigates the Ore condition for the group ring of R. Thompson's group F, reducing the problem to homogeneous elements, and explores solutions for low degrees, linking the problem to the group's amenability.

Contribution

It reduces the Ore condition problem for the group ring of F to homogeneous elements and analyzes specific cases for degrees 1 and 2, providing new insights into the amenability question.

Findings

Ore condition holds for degree 1 elements.

Ore condition holds for certain degree 2 linear combinations.

Open problems remain for higher degrees and larger generating sets.

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 $F$.