On solvability of certain equations of arbitrary length over torsion-free groups

Abstract

Let $G$ be a non-trivial torsion free group and $s(t)=g_1{t}^{{\epsilon }_1}{g}_2{t}^{{\epsilon }_2}\cdots g_n{t}^{{\epsilon }_n}=1(g_i\in G,{\epsilon }_i=\pm 1)$ be an equation over $G$ containing no blocks of the form $t^{-1}{g}_i{t}^{-1},g_i\in G$. In this paper we show that $s(t)=1$ has a solution over $G$ provided a single relation on coefficients of $s(t)$ holds. We also generalize our results to equations containing higher powers of $t$. The later equations are also related to Kaplansky zero-divisor conjecture \cite{K}.