Systems of equations over the group ring of Thompson’s group F
Victor Sergeevich Guba · Communications in Algebra · 2022
Let R=K[G] 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 a,b∈R there exist u,v∈R such that au = bv, where u≠0 or v≠0. 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 Y⊂F with the property |AY|<43|Y|, where A={x0,x1,x2}. This implies some result on the systems of equations. We show that for any element b in the group ring of F, the equation (1−x0)u=bv has a non-zero solution. The corresponding fact for 1−x1 instead of 1−x0 remains open. We deduce that for any m≥1 the system (1−x0)u0=(1−x1)u1=⋯=(1−xm)um has nonzero solutions in the group ring of F. We also analyze the equation (1−x0)u=(1−x1)v giving a precise explicit description of all its solutions in K[F]. This is important since to any group relation between x0, x1 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.