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Only their cardinality can prevent groups from being quotients of weakly branch groups

  • J. Moritz Petschick
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Abstract

It is a well-known fact that every proper quotient of a branch group is a virtually abelian group. We record that every group of cardinality at most 2ℵ0 is a quotient of a weakly branch group; in fact, there are weakly branch groups that are split extensions with free complements. Furthermore, if a group is generated by d elements, there is a weakly branch group generated by (d+2) elements mapping onto it. As applications of this fact, we resolve a question of Thillaisundaram on the existence of weakly branch but non-branch groups with maximal subgroups of infinite index and, partially, a question of Cox and Thillaisundaram on the existence of weakly branch groups that are not invariably generated.

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Cite

BibTeXRIS
BibTeX
@misc{petschick2026only,
  title = {{Only their cardinality can prevent groups from being quotients of weakly branch groups}},
  author = {J. Moritz Petschick},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21727},
  eprint = {2610.21727},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21727},
}
RIS
TY  - UNPB
TI  - Only their cardinality can prevent groups from being quotients of weakly branch groups
AU  - J. Moritz Petschick
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21727
UR  - https://arxiv.org/abs/2610.21727
AB  - It is a well-known fact that every proper quotient of a branch group is a virtually abelian group. We record that every group of cardinality at most 2^ℵ_0 is a quotient of a weakly branch group; in fact, there are weakly branch groups that are split extensions with free complements. Furthermore, if a group is generated by d elements, there is a weakly branch group generated by (d+2) elements mapping onto it. As applications of this fact, we resolve a question of Thillaisundaram on the existence of weakly branch but non-branch groups with maximal subgroups of infinite index and, partially, a question of Cox and Thillaisundaram on the existence of weakly branch groups that are not invariably generated.
ER  -