Systems of equations over the group ring of Thompson’s group <i>F</i>

Let R=K[G] be a group ring of G over field K. It is known that if amenable then R satisfies the Ore condition: for any a,b∈R there exist u,v∈R such au = bv, where u≠0 or v≠0. also true groups non-zero solution exists finite system linear equations R, number unknowns exceeds equations. Recently Bartholdi proved converse. As consequence this theorem, Kielak R. Thompson’s F and only it condition. The amenability problem long-standing open question.In article, we prove some their systems have solutions in rings F. We improve results by Donnelly showing sets Y⊂F with property |AY|<43|Y|, A={x0,x1,x2}. This implies result on show element b F, equation (1−x0)u=bv has solution. corresponding fact 1−x1 instead 1−x0 remains open. deduce m≥1 (1−x0)u0=(1−x1)u1=⋯=(1−xm)um nonzero analyze (1−x0)u=(1−x1)v giving precise explicit description all its K[F]. important since to relation between x0, x1 one can naturally assign So help estimate relations given length generators.