Solutions of equations over \(\omega\)-nilpotent groups (Q1100557)

This paper deals with the uniqueness of solutions of equations over \(\omega\)-nilpotent groups. An \(\omega\)-nilpotent group is also known as a residually nilpotent group i.e. a group whose terms of the lower central series have trivial intersection. If \({\mathbb{Z}}[G]\) is the integral group ring of the group G, then \({\mathbb{Z}}[G]^{\wedge}\) denotes the completion of \({\mathbb{Z}}[G]\) in the \(I_ G\)-adic topology, where \(I_ G\) is the augmentation ideal. Let \(U_ 1({\mathbb{Z}}[G]^{\wedge})\) denote the group of units \(\equiv 1 mod I_ G\) of \({\mathbb{Z}}[G]\). An equation over G is given by an element w(t) of the free product \(G*<t>.\) One of the main results of the paper is that if G is finitely generated and \(\omega\)-nilpotent, then the canonical homomorphism \(f: G\to U_ 1({\mathbb{Z}}[G]^{\wedge})\) with \(g\mapsto 1+(g-1)\), \(g-1\in I_ G\) is an injection and if \(w=w(t)\in G*<t>\) with \(\ell_ t(w)=\pm 1\) (where \(\ell_ t(w)\) denotes the exponent sum of t in w), then the equation \(w(t)=1\) has a unique solution in \(U_ 1({\mathbb{Z}}[G]^{\wedge})\). Here G is identified with its image under the injection f.