Tame almost primitive elements
Let \(G\) be a finitely generated group, \(V\) a set of laws, and \(\mathcal V\) the variety defined by \(V\). An element \(\omega\in G\) is \(\mathcal V\)-generic in \(G\) if \(\omega\in V(G)\) but \(\omega\notin V(K)\) for any proper subgroup \(K<G\), primitive if it is contained in some minimal generating set for \(G\), almost primitive if it is primitive in every proper subgroup of \(G\) which contains it, and tamely almost primitive if it is almost primitive, and whenever any power \(\omega^\alpha\in H\leq G\) where \(\alpha\) is positive and minimal and \(H\) is finitely-generated, then either \(\omega^\alpha\) is primitive in \(H\) or the index of \(H\) in \(G\) is \(\alpha\). The first result of the paper shows that \(a^\alpha b^\beta\) is tame almost primitive in the free group \(F_2=\langle a,b\rangle\) if and only if \(\alpha=\beta=2\). The remaining sections of the paper are devoted to the study of generic, almost primitive and tame almost primitive elements in surface groups. Let \(G=\langle a_1,\dots,a_n;\;[a_1,a_2]\cdots[a_{n-1},a_n]=1\rangle\), \(n\geq 4\), \(n\) even. It is shown that \(\omega=a_1^{p_1}\cdots a_n^{p_n}\), where all the \(p_i\) are primes, is almost primitive, but it is tamely almost primitive if and only if all the \(p_i\) are equal to 2.
scientific article; zbMATH DE number 1498582
scientific article