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user6976
user6976

It is not true. A counterexample was constructed in Ashmanov, I. S.; Olʹshanskiĭ, A. Yu. Abelian and central extensions of aspherical groups. (Russian) Izv. Vyssh. Uchebn. Zaved. Mat. 1985, no. 11, 48–60, 85. They even contsructed a noetherian group $G$ and a word $v$ with finite verbal subgroup $v(G)$ and marginal subgroup $v^*(G)$ of infinite index (that answered a question of Ph. Hall).

It is not true. A counterexample was constructed in Ashmanov, I. S.; Olʹshanskiĭ, A. Yu. Abelian and central extensions of aspherical groups. (Russian) Izv. Vyssh. Uchebn. Zaved. Mat. 1985, no. 11, 48–60, 85. They even contsructed a noetherian group $G$ and a word $v$ with finite verbal subgroup $v(G)$ and marginal subgroup $v^*(G)$ of infinite index (that answered a question of Ph. Hall).

It is not true. A counterexample was constructed in Ashmanov, I. S.; Olʹshanskiĭ, A. Yu. Abelian and central extensions of aspherical groups. Izv. Vyssh. Uchebn. Zaved. Mat. 1985, no. 11, 48–60, 85. They even contsructed a noetherian group $G$ and a word $v$ with finite verbal subgroup $v(G)$ and marginal subgroup $v^*(G)$ of infinite index (that answered a question of Ph. Hall).

Source Link
user6976
user6976

It is not true. A counterexample was constructed in Ashmanov, I. S.; Olʹshanskiĭ, A. Yu. Abelian and central extensions of aspherical groups. (Russian) Izv. Vyssh. Uchebn. Zaved. Mat. 1985, no. 11, 48–60, 85. They even contsructed a noetherian group $G$ and a word $v$ with finite verbal subgroup $v(G)$ and marginal subgroup $v^*(G)$ of infinite index (that answered a question of Ph. Hall).