How to generalize known results on equations over groups

Abstract

The known facts about solvability of equations over groups are considered from a more general point of view. A generalized version of the theorem about solvability of unimodular equations over torsion-free groups is proved. In a special case, this generalized version become a multivariable variant of this theorem. For unimodular equations over torsion free groups, we prove an analogue of Magnus's Freiheitssatz, which asserts that there exists a solution with good behavior with respect to free factors of the initial group.

9 pages. V.4: minor corrections in the proof of the main theorem (p.5). Russian version of this paper is at http://mech.math.msu.su/department/algebra/staff/klyachko/papers.htm