On the Boffa alternative (Q2717735)

The group law \(u=v\) is called positive if \(u,v\) are words of the free semigroup \(F\) generated by \(x_1,x_2,\dots\). Let \(G^*\) denote a nonprincipal ultrapower of a group \(G\). The authors say that a group \(G\) is an \(F^*\)-group if \(G^*\) contains a free nonabelian subsemigroup.NEWLINENEWLINENEWLINEQuestion (Q) of \textit{M. Boffa} [in Model theory of groups and automorphism groups, Lond. Math. Soc. Lect. Note Ser. 224, 134-143 (1997; Zbl 0945.03053)] suggests the following alternative: a group either satisfies a positive law or is an \(F^*\)-group. In the paper the large class \(\mathcal C\) of groups satisfying the Boffa alternative is presented. To recall the definition of the class \(\mathcal C\) introduced by \textit{R. G. Burns, O. Macedońska} and \textit{Yu. Medvedev} [in J. Algebra 195, No. 2, 510-525 (1997; Zbl 0886.20022)], we denote by \(\Delta_1\) the class of groups contained in finite products of varieties \({\mathcal V}_1\cdots{\mathcal V}_m\), where \({\mathcal V}_i\) is either the variety \({\mathcal A}^n\) of all soluble groups of solubility class \(n\), or the restricted Burnside variety \({\mathcal B}_k\) (that is, the variety generated by all finite groups of exponent \(k\)) for various \(n,k\). Now we define inductively: \(\Delta_{n+1}=\{\)groups locally in \(\Delta_n\}\cup\{\)groups residually in \(\Delta_n\}\). The class \(\mathcal C\) is defined as the union \({\mathcal C}=\bigcup_n\Delta_n\).NEWLINENEWLINENEWLINEThe main result of the paper is the following Theorem 1. If \(G\) is in the class \(\mathcal C\) then either \(G\) satisfies a positive law or \(G\) is an \(F^*\)-group.