High Rank and Multiplicity in Random and Perfect Profinite Groups

Abstract

We prove that almost sure topological finite generation holds for a general class of random models of profinite groups. We deduce that in the models introduced by Liu--Wood and Sawin--Wood, one has that finite presentation holds almost surely, with almost sure control of the deficiency in the presentation. In particular this settles questions raised in work of Liu--Wood and Sawin--Wood. Using the same underlying principle, we show that there exists a unique universal d-generated perfect profinite group: its finite quotients are precisely the finite d-generated perfect groups. This settles a conjecture of Nikolov. We show in addition that this group is projective and admits a profinite presentation with d generators and d relations, and with no fewer relations on d generators. The main underlying theme is to bring in an insight from crown theory: groups with high rank are always witnessed by a crown with large multiplicity. This approach was discovered independently by ChatGPT5.5 pro and Aletheia, an internal agent at Google DeepMind.

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BibTeX RIS

BibTeX

@misc{pagano2026high,
  title = {{High Rank and Multiplicity in Random and Perfect Profinite Groups}},
  author = {Carlo Pagano and Mark Shusterman},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.30200},
  eprint = {2609.30200},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30200},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - High Rank and Multiplicity in Random and Perfect Profinite Groups
AU  - Carlo Pagano
AU  - Mark Shusterman
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.30200
UR  - https://arxiv.org/abs/2609.30200
AB  - We prove that almost sure topological finite generation holds for a general class of random models of profinite groups. We deduce that in the models introduced by Liu--Wood and Sawin--Wood, one has that finite presentation holds almost surely, with almost sure control of the deficiency in the presentation. In particular this settles questions raised in work of Liu--Wood and Sawin--Wood. Using the same underlying principle, we show that there exists a unique universal d-generated perfect profinite group: its finite quotients are precisely the finite d-generated perfect groups. This settles a conjecture of Nikolov. We show in addition that this group is projective and admits a profinite presentation with d generators and d relations, and with no fewer relations on d generators. The main underlying theme is to bring in an insight from crown theory: groups with high rank are always witnessed by a crown with large multiplicity. This approach was discovered independently by ChatGPT5.5 pro and Aletheia, an internal agent at Google DeepMind.
ER  -

Abstract

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