On the residual polynilpotence of tree products of groups with locally cyclic edge subgroups

Abstract

Let 𝒩 be a non-empty class of nilpotent groups closed under taking subgroups, quotient groups, finite direct products, and unrestricted direct powers of the form ∏_{x ∈ X}Y_x, where X ∈ 𝒩 and Y_x (x ∈ X) are isomorphic copies of a group Y ∈ 𝒩. Let also the symbols 𝒩^tf and Poly_k𝒩^tf denote the class of all torsion-free 𝒩-groups and the class of polynilpotent groups such that X ∈ Poly_k𝒩^tf if and only if X has a subnormal series of length at most k whose factors belong to 𝒩^tf. We prove that if all free groups are residually 𝒩^tf-groups and 2 leqslant n < ∞, then any tree product of n residually 𝒩^tf-groups with non-trivial torsion-free locally cyclic edge subgroups is residually a (Poly_n+1𝒩^tf)-group. In particular, all cyclically-pinched one-relator groups are residually finitely generated (Poly₃𝒩̅^tf)-groups, where 𝒩̅^tf is the class of all torsion-free nilpotent groups. We prove also that any tree product of n 𝒩-groups with non-trivial torsion-free locally cyclic edge subgroups has a homomorphism onto a (Poly_n−1𝒩^tf)-group that is injective on all edge subgroups.

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BibTeX RIS

BibTeX

@misc{sokolov2026residual,
  title = {{On the residual polynilpotence of tree products of groups with locally cyclic edge subgroups}},
  author = {E. V. Sokolov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23126},
  eprint = {2610.23126},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23126},
}

RIS

TY  - UNPB
TI  - On the residual polynilpotence of tree products of groups with locally cyclic edge subgroups
AU  - E. V. Sokolov
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.23126
UR  - https://arxiv.org/abs/2610.23126
AB  - Let 𝒩 be a non-empty class of nilpotent groups closed under taking subgroups, quotient groups, finite direct products, and unrestricted direct powers of the form ∏_x ∈ XY_x, where X ∈ 𝒩 and Y_x (x ∈ X) are isomorphic copies of a group Y ∈ 𝒩. Let also the symbols 𝒩^tf and Poly_k𝒩^tf denote the class of all torsion-free 𝒩-groups and the class of polynilpotent groups such that X ∈ Poly_k𝒩^tf if and only if X has a subnormal series of length at most k whose factors belong to 𝒩^tf. We prove that if all free groups are residually 𝒩^tf-groups and 2 leqslant n < ∞, then any tree product of n residually 𝒩^tf-groups with non-trivial torsion-free locally cyclic edge subgroups is residually a (Poly_n+1𝒩^tf)-group. In particular, all cyclically-pinched one-relator groups are residually finitely generated (Poly_3𝒩̅^tf)-groups, where 𝒩̅^tf is the class of all torsion-free nilpotent groups. We prove also that any tree product of n 𝒩-groups with non-trivial torsion-free locally cyclic edge subgroups has a homomorphism onto a (Poly_n−1𝒩^tf)-group that is injective on all edge subgroups.
ER  -

Abstract

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