ON SOLVABILITY OF CERTAIN EQUATIONS OF ARBITRARY LENGTH OVER TORSION-FREE GROUPS

Let G be a nontrivial torsion-free group and s\left( t \right) = {g_1}{t^{{\varepsilon _1}}}{g_2}{t^{{\varepsilon _2}}} \ldots {g_n}{t^{{\varepsilon _n}}} = 1\left( {{g_i} \in G,{\varepsilon_i} = \pm 1} \right)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\left( t \right) = 1has a solution over G provided a single relation on coefficients of s ( t ) holds. We...