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Generic elements in certain groups. (English) Zbl 0867.20025

Kim, A. C. (ed.) et al., Groups - Korea ’94. Proceedings of the international conference, Pusan, Korea, August 18-25, 1994. Berlin: Walter de Gruyter. 289-294 (1995).
The commutator \([a,b]=aba^{-1}b^{-1}\) of the free group \(F\) on \(\{a,b\}\) has the interesting property that if \(\alpha:G\to F\) is any homomorphism which takes an element of the commutator subgroup of \(G\) to \([a,b]\), then \(\alpha\) is surjective. This paper looks at a generalization of this property. Definition. An element \(h\) of a group \(H\) is said to be generic in \(H\) with respect to the verbal subgroup \(\mathcal V\), when \(h\in{\mathcal V}\) and, for every group \(G\) and every homomorphism \(\phi:G\to H\), if there exists \(g\in G\) such that \(\phi(g)=h\), then \(\phi\) is surjective. (Note that the definition of verbal subgroup used in this paper is somewhat more general that the one in common use, however, it amounts to the same thing for countable groups.) The verbal subgroups considered are the subgroups \({\mathcal V}_n\) determined by the word \(z^nxyx^{-1}y^{-1}\) (so that \({\mathcal V}_0\) is just the commutator group). A sufficient condition for a word to be generic with respect to \({\mathcal V}_k\) was given by A. Dold [Arch. Math. 50, No. 6, 564-569 (1988; Zbl 0628.20026)], and this result is used to show that there is an algorithm to determine whether any element of a finitely-generated free group is generic with respect to \(\mathcal V\). The paper also raises the question of the existence of generic elements in groups other than free groups. Perhaps something close to free groups, like the “hyperbolic groups” defined by M. Gromov [in: Essays in group theory, Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)] would be a good place to start.
For the entire collection see [Zbl 0857.00028].

MSC:

20E10 Quasivarieties and varieties of groups
20F65 Geometric group theory
20E05 Free nonabelian groups
20F12 Commutator calculus
20E36 Automorphisms of infinite groups
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