### Systems of equations over the group ring of Thompson's group F [article]

Victor Guba
2022 arXiv   pre-print
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 u0 or v0. 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
more » ... if and only if it satisfies the Ore condition. The amenability problem for F is a long-standing open question. In this paper 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| < 4/3|Y|, where A={x_0,x_1,x_2}. 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-x_0)u=bv has a non-zero solution. The corresponding fact for 1-x_1 instead of 1-x_0 remains open. We deduce that for any m≥1 the system (1-x_0)u_0=(1-x_1)u_1=⋯=(1-x_m)u_m has nonzero solutions in the group ring of F. We also analyze the equation (1-x_0)u=(1-x_1)v giving a precise explicit description of all its solutions in K[F]. 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.