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PreprintCC BY 4.0arXiv

The word problem in verbal products of groups

  • Javier Brude
  • Román Sasyk
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Abstract

We prove that nilpotent and metabelian products of groups with decidable word problem have decidable word problem. In contrast, we exhibit families of groups with decidable word problem for which their solvable product have undecidable word problem. In doing so, we exhibit hyperbolic groups with undecidable membership problem in its derived subgroups.

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Cite

BibTeXRIS
BibTeX
@misc{brude2026word,
  title = {{The word problem in verbal products of groups}},
  author = {Javier Brude and Román Sasyk},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.31289},
  eprint = {2609.31289},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.31289},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The word problem in verbal products of groups
AU  - Javier Brude
AU  - Román Sasyk
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.31289
UR  - https://arxiv.org/abs/2609.31289
AB  - We prove that nilpotent and metabelian products of groups with decidable word problem have decidable word problem. In contrast, we exhibit families of groups with decidable word problem for which their solvable product have undecidable word problem. In doing so, we exhibit hyperbolic groups with undecidable membership problem in its derived subgroups.
ER  -