On Andrews–Curtis conjectures for soluble groups

Abstract

The Andrews–Curtis conjecture claims that every normally generating [Formula: see text]-tuple of a free group [Formula: see text] of rank [Formula: see text] can be reduced to a basis by means of Nielsen transformations and arbitrary conjugations. Replacing [Formula: see text] by an arbitrary finitely generated group yields natural generalizations whose study may help disprove the original and unsettled conjecture. We prove that every finitely generated soluble group satisfies the generalized Andrews–Curtis conjecture in the sense of Borovik, Lubotzky and Myasnikov. In contrast, we show that some soluble Baumslag–Solitar groups do not satisfy the generalized Andrews–Curtis conjecture in the sense of Burns and Macedońska.