On the Andrews-Curtis groups: non-finite presentability

Abstract

The Andrews-Curtis conjecture remains one of the outstanding open problems in combinatorial group theory. It claims that every normally generating $r$-tuple of a free group $F_r$ of rank $r\geq 2$ can be reduced to a basis by means of Nielsen transformations and arbitrary conjugations. These transformations generate the so-called Andrews-Curtis group AC($F_r$). The groups AC($F_r$) ($r = 2, 3, \ldots$) are actively investigated and allows various generalizations, for which there are a number of results. At the same time, almost nothing is known about the structure and properties of the original groups AC($F_r$). In this paper we define a class $\{A_{r, s}: r, s \geq 1\}$ of generalized Andrews-Curtis groups in which any group $A_{r,r}$ is isomorphic to the Andrews-Curtis group AC($F_r$). We prove that every group $A_{2,s}$\, ($s \geq 1$) is non-finitely presented. Hence the Andrews-Curtis group AC($F_2$) $\simeq A_{2,2}$ is non-finitely presented. Thus, we give a partial answer to the well-known question about the finite presentability of the groups AC($F_r$), explicitly stated by J. Swan and A. Lisitsa in the Kourovka notebook \cite{KN} (Question 18.89).