An asphericity conjecture and Kaplansky problem on zero divisors (Q1295979)

Let \(\langle X,R\rangle\) be a group presentation; it is said to be aspherical provided its geometric realization is aspherical. Given a presentation \(\mathcal P=\langle X,R\rangle\), the author considers a presentation \(\mathcal P'\) obtained from \(\mathcal P\) by adding a new generator \(y\) (say) and a new relator which as a word in the generators of \(\mathcal P'\) involves \(y\) with non-zero exponent sum; he supposes that \(\mathcal P\) is aspherical and that the group \(H\) corresponding to \(\mathcal P\) embeds into the group \(G\) corresponding to \(\mathcal P'\). Under these circumstances, the author conjectures that \(\mathcal P'\) is aspherical if and only if \(G\) is torsion free. He proves that, if \(\mathcal P'\) is a counterexample to the conjecture, the group \(G\) is torsion free with integral group ring \(\mathbb{Z} G\) having a non-trivial zero divisor. He also lists a number of cases where he proves the conjecture to be true.