On Andrews-Curtis conjectures for soluble groups

TL;DR

This paper investigates the Andrews-Curtis conjecture within finitely generated soluble groups, proving it holds for some but not all such groups, thus advancing understanding of the conjecture's scope.

Contribution

It proves the generalized Andrews-Curtis conjecture for all finitely generated soluble groups and provides counterexamples among soluble Baumslag-Solitar groups.

Findings

Finitely generated soluble groups satisfy the generalized Andrews-Curtis conjecture.

Some soluble Baumslag-Solitar groups do not satisfy the conjecture.

The results differentiate between classes of soluble groups regarding the conjecture.

Abstract

The Andrews-Curtis conjecture claims that every normally generating $n$-tuple of a free group $F_n$ of rank $n \ge 2$ can be reduced to a basis by means of Nielsen transformations and arbitrary conjugations. Replacing $F_n$ 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 Macedo\'nska.