On Solvability of Certain Equations of Arbitrary Length Over Torsion-Free Groups

Abstract

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 $s\left( t \right) = 1$ 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.

Yet another Freiheitssatz: Mating finite groups with locally indicable ones

“…The study of this notion has a long history, see, e.g., [1, 2, 4, 6–12, 15–18, 20–25, 27, 28, 32], and references therein. The following result is well known.…”

On certain equations of arbitrary length over torsion-free groups

“…The method depends on application of either weight test or the curvature distribution method [10], to each case of an equation. In [12], it was proved that certain equations of arbitrary length have a solution over torsion free groups provided a single relation among elements of G holds.…”