An Asphericity Conjecture and Kaplansky Problem on Zero Divisors

Suppose a group representationH = is aspherical,x ,W( x) is a word in alphabet ( x) ± 1with nonzero sum of exponents onx, and the groupHnaturally embeds inG = x W( x) . It is conjectured that the presentationG = x W( x) is aspherical if and only ifGis torsion free. It is proven that if this conecture is false andG = x W( x) is a counterexample, then the integral group ring (G) of torsion free groupGwill contain zero divisors. Some special cases when this conjecture holds are also indicated.