Recalcitrance in groups II

Abstract

Abstract. Motivated by the well-known conjecture of Andrews and Curtis [1], we consider the question of how, in a given n-generator group G, any ordered n-tuple of ''annihilators'' of G, that is, with normal closure all of G, can be transformed by standard moves into a generating n-tuple. The recalcitrance of G is defined to be the least number of elementary standard moves (''elementary M-transformations'') by means of which every annihilating n-tuple can be transformed into a generating n-tuple. We obtain upper estimates for the recalcitrance of n-generator finite groups-thus quantifying a result from [2]-and of a wide class of n-generator solvable groups, thus extending and correcting a result from [3].

On Andrews–Curtis conjectures for soluble groups

“…Our main results are Theorem 1.3, Theorem 1.9 and Corollary 1.10 below. Theorem 1.3 elaborates on results of [20,8] to settle the generalized Andrews conjecture in the sense of [4] for soluble groups. Theorem 1.9 and Corollary 1.10 solve [8, Open Problem] and show in particular that the metabelian Baumslag-Solitar group a, b aba −1 = b 11 does not satisfy the generalized Andrews conjecture in the sense of [7].…”

“…By [8, Theorem 3.1], the recalcitrance of a soluble group G of rank n is bounded by 2n − 1 if its abelianization homomorphism is coessential. The authors ask in [8,Open Problem] whether the latter condition is necessary. We answer this question in the positive and exhibit subsequently a two-generated metabelian group with infinite recalcitrance.…”

“…We answer this question in the positive and exhibit subsequently a two-generated metabelian group with infinite recalcitrance. Theorem 1.9 (Lemma 3.1 and Theorem 3.1 of [8]). Let G be a finitely generated soluble group of rank n. Then G satisfies GACC 2 if and only if its abelianization homomorphism is coessential.…”

“…3. The Andrews-Curtis conjecture in the sense of [8] This section is dedicated to the proofs of Theorem 1.9 and Corollary 1.10. We shall use the following definition in a preliminary result.…”

Coessential Abelianization Morphisms in the Category of Groups

“…This question arose in the course of our quantitative investigation [2,6] of the Andrews-Curtis problem for an arbitrary group G of rank n, whether every normal generating n-tuple (h 1 , h 2 , . .…”