Systems of equations over the group ring of Thompson’s group F

Abstract

Let be a group ring of a group G over a field K. It is known that if G is amenable then R satisfies the Ore condition: for any there exist such that au = bv, where or It is also true for amenable groups that a non-zero solution exists for any finite system of linear equations over R, where the number of unknowns exceeds the number of equations. Recently Bartholdi proved the converse. As a consequence of this theorem, Kielak proved that R. Thompson’s group F is amenable if and only if it satisfies the Ore condition. The amenability problem for F is a long-standing open question. In this article, we prove that some equations or their systems have non-zero solutions in the group rings of F. We improve some results by Donnelly showing that there exist finite sets with the property where This implies some result on the systems of equations. We show that for any element b in the group ring of F, the equation has a non-zero solution. The corresponding fact for instead of remains open. We deduce that for any the system has nonzero solutions in the group ring of F. We also analyze the equation giving a precise explicit description of all its solutions in This is important since to any group relation between x 0, x 1 in F one can naturally assign such a solution. So this can help to estimate the number of relations of a given length between generators.