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Texas A&M University
Mathematics

Groups and Dynamics Seminar

Date: October 8, 2020

Time: 12:00PM - 1:00PM

Location: online

Speaker: Victor Guba, Vologda State University

  

Title: 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 $s,t\in R$ such that $as=bt$ and $s\ne0$ or $t\ne0$. 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 same group. It is easy to see that the problem on the Ore condition is equivalent to the same question for the monoid ring $K[M]$, where $M$ is the monoid of positive elements of $F$. In this paper we reduce the problem for 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 minimal solutions of the Ore equation. For the case of degree $2$, we study the case of linear combinations of monomials from $S=\{x_0^2,x_0x_1,x_0x_2,x_1^2,x_1x_2\}$. We show that this set is not doubling, that is, there are nonempty finite subsets $X\subset M\subset F$ such that $|SX| < 2|X|$. In particular, Ore condition holds for linear combinations of these monomials. The case of monomials of higher degree is open as well as the case of degree $2$ for the monomials on $x_0,x_1,...,x_m$, where $m\ge3$. Recall that negative answer for one of these cases will immediately imply non-amenability of $F$.