Generating systems of groups and Reidemeister-Whitehead torsion
Two (minimal) systems of generators of a group are called Nielsen equivalent, if the change of generators can be lifted to the free groups generated by them. The purpose of the paper is to define a generally applicable invariant which can be used to show that a pair of generating systems is not Nielsen equivalent. For a finitely generated group \(G\) an abelian group \({\mathcal N}(G)\) is defined and to a pair \(x\), \(y\) of minimal generating systems an element \({\mathcal N}(y,x) \in {\mathcal N}(G)\) is assigned which only depends on the Nielsen classes of \(x\) and \(y\), and which vanishes, if \(x\) and \(y\) are Nielsen equivalent. \({\mathcal N}(G)\) is the Whitehead group of the quotient of \(\mathbb{Z} G\) modulo the ``Fox ideal'' which is generated by the Fox derivatives of the defining relators of \(G\) by the generators \(x\) of \(G\). To make the invariant computable a homomorphism of \(\mathbb{Z} G\) into a matrix ring over a unitary commutative ring is employed, and the determinant of the image of the Jacobian of the two generating systems \(x\) and \(y\) is considered. If this determinant is not of a certain ``trivial-unit-type'', then \(x\) and \(y\) are not Nielsen equivalent. The authors prove functorial properties of \(\mathcal N\), and consider the behaviour of their invariant under free products. The last paragraph is devoted to applications: finite abelian groups are considered, and known results recovered and extended; knot groups, especially of 2- and certain 3-bridge knots, are treated. These examples can be used to exhibit Nielsen inequivalent generators for 3-manifolds obtained by Dehn surgery on \(S^ 3\) along such knots. This implies non-isotopy of associated Heegaard splittings of this manifold.
scientific article; zbMATH DE number 202989