Geodesic
Rationality of verbal subsets in solvable groups
Algebra i logika, Tome 57 (2018) no. 1, pp. 57-72

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A verbal subset of a group $G$ is a set $w[G]$ of all values of a group word $w$ in this group. We consider the question whether verbal subsets of solvable groups are rational in the sense of formal language theory. It is proved that every verbal subset $w[N]$ of a finitely generated nilpotent group $N$ with respect to a word w with positive exponent is rational. Also we point out examples of verbal subsets of finitely generated metabelian groups that are not rational.
Mots-clés : solvable group
Keywords: verbal subset, verbal subgroup, rational set, formal language. 
@article{AL_2018_57_1_a3,
     author = {V. A. Roman'kov},
     title = {Rationality of verbal subsets in solvable groups},
     journal = {Algebra i logika},
     pages = {57--72},
     publisher = {mathdoc},
     volume = {57},
     number = {1},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2018_57_1_a3/}
}
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V. A. Roman'kov. Rationality of verbal subsets in solvable groups. Algebra i logika, Tome 57 (2018) no. 1, pp. 57-72. http://geodesic.mathdoc.fr/item/AL_2018_57_1_a3/