The Andrews-Curtis conjecture and the recalcitrance of groups

In 1924, Nielsen defined certain automorphisms of the free group F of rank n, and showed that they generate the automorphism group of F [29]. These "elementary" automorphisms are given by so-called "elementary Nielsen transformations" of generating n-tuples of F; Nielsen showed that any two generating n-tuples of F are transformable into one another by means of a finite sequence of such transformations.;In this work we focus on the algebraic aspect of the conjecture, and on its generalization to the class of finitely generated groups. Thus, we consider the question as to whether the annihilating n-tuples of a group of rank n can be transformed by means of extended Nielsen transformations into generating n-tuples. We begin by investigating the Nielsen classes of abelian groups. We generalize the classification of [14] from finite to finitely generated abelian groups, then use this result to establish a property of the abelianization homomorphism of certain finitely generated groups, relating the Nielsen classes of the abelianized group to the AC-classes of the group.;In 1993, Burns and Macedonska introduced so-called "M-transformations", and showed that they are, in a certain specific sense, equivalent to extended Nielsen transformations [10]. The supremum of the numbers of M-transformations needed to transform annihilating n-tuples of a group G of rank n into generating n-tuples is called the recalcitrance of G, as introduced in [9].;Our main results give upper bounds on the recalcitrance of several classes of groups, chiefly solvable groups, finite groups, and finitely generated groups with a principal series. We also investigate certain possible counterexamples to the AC-conjecture, bound the recalcitrance of a family of annihilating pairs in a free group of rank 2 from below, and consider so-called "weak" AC-equivalence in this context. Finally, we examine briefly the connections between the group-theoretic and topological aspects of the Andrews-Curtis conjecture.;In 1965, Andrews and Curtis introduced the extended Nielsen transformations , obtained by supplementing the elementary Nielsen transformations by conjugations [3, 4]. They conjectured that any two "annihilating" n-tuples of F are equivalent under the extended transformations. (An n-tuple of F is annihilating or normal generating if the normal subgroup it generates in F is all of F.)...