Systems of equations over locally \(p\)-indicable groups
It is conjectured that any independent system of equations over any group \(G\) has a solution in some group containing \(G\). Howie has shown the conjecture to be true in the case of locally indicable groups. His proof relies on methods from low dimensional topology. The corresponding result in the case of locally \(p\)-indicable groups is due to Gersten. The paper under review provides two new proofs of Gersten's result. The first proof is combinatorial. The main technical tool is the Reidemeister-Schreier rewriting process. Howie's tower construction appears in algebraic disguise. The second proof provides a particular group over \(G\) containing the solutions of any \(p\)-independent system of equations. Let \(N\rightarrowtail F\twoheadrightarrow G\) be a free presentation of a finitely generated locally \(p\)-indicable group \(G\). Let \(\hat F\) denote the pro-\(p\)-completion of \(F\) and \(N^*\) the normal closure of \(N\) in the discrete group \(\hat F\). If \(G\) is locally \(p\)-indicable then \(N=F\cap N^*\) and the quotient \(\hat F/N^*\) is shown to contain the solutions of any \(p\)-independent system of equations over \(G\). This situation is circumscribed by the term ``\(p\)-adic solution'' by the author. However it has to be pointed out that ``\(p\)-adic solutions'' live in a discrete group rather than in the \(p\)-adic completion of \(G\). Moreover the author uses Stallings' notation \(C(p)\), originally referring to a property of free groups, in a different sense. In this paper \(C(p)\) is a property of \(G\) respectively of its free presentation \(G=F/N\). This distinction has to be kept in mind when interpreting the remark: ``Restated in Stallings' terminology, this Theorem says that locally \(p\)-indicable groups satisfy \(C(p)\),..'' Using Stallings' original definition (which refers to free groups only), Howie showed that conjecture \(C(p)\) in the sense of Stallings does not hold.
finitely generated locally \(p\)-indicable group
scientific article; zbMATH DE number 3918658