ON SMALL LENGTH EQUATIONS OVER TORSION-FREE GROUPS

Abstract

Let G be a group, <t> the free group generated by t and let r(t)∈G*<t>. The equation r(t)=1 is said to have a solution over G if it has a solution in some group that contains G. There is a conjecture (attributed to F. Levin) that if G is a torsion-free group, then any equation has a solution over G. In this paper we verify this conjecture in the case when the sum of the absolute values of the exponent of t in r(t) is not greater than six.