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Results · Groups and their finite shadows

Amenable groups with nearly exponential sofic profile, and quantum channels that need nearly linear memory

Seth Douglas and Nidhal Mghirbi · version 1.2.0 · Zenodo, · 78 pages · arXiv:2610.04271 · doi:10.5281/zenodo.23875973 · PDF

In plain wordsA finitely presented amenable group can have a finite piece that permutations imitate only with nearly exponentially many points; it yields an explicit quantum channel that, applied n times by a device releasing each output at once, needs memory times purity of about n.

The result

As the paper states it, in briefA finitely presented elementary amenable group with a finite piece whose sofic profile is exp⁡(r1+o(1)).

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Novelty and credit

Novelty
New, with named precedent: Musat and Rørdam gave channels in the closure with no finite tracial factorization, and Slofstra a finitely presented group with profile at least exp(rc) that is not known to be sofic. It extends the authors’ Houghton paper. New, in the paper’s words: the quantitative profile, with exponent 1 − o(1).

Versionsseven, each with its own DOI

Every archived version, newest first. Dates are Zenodo publication dates. All versions: 10.5281/zenodo.23050302, which always resolves to the newest.
VersionDateDOINotes
1.2.010.5281/zenodo.23875973Adds a sweep form of the counting criterion; for the dyadic lamplighter of Section 8.1 the lower bound rises from exp(c r1/3) to exp(c r1/2). The discussion of Thompson’s groups cites the recent soficity preprint of Chaudkhari, Juschenko and Muliarchyk (arXiv:2610.25415), on which the soficity of the subgroup L relies. PDF hosted here.
1.1.210.5281/zenodo.23111927Correction: Credit added in the introduction: Gromov and Weiss; OpenAI’s announcement, Fournier-Facio, and Kun and Thom on non-sofic groups; Slofstra and Vidick, and Slofstra; Arzhantseva and Cherix; and work on channel simulation, memory costs and collision models.
1.1.110.5281/zenodo.23108312Correction: Precision: a step in an appendix’s Choi-rank count is corrected (the bound is unchanged), and a set of coefficients is corrected from {±1, ±2^−1/2} to {1, ±2^−1/2}.
1.1.010.5281/zenodo.23107117
1.0.210.5281/zenodo.23070871Correction: Credit made precise: Cornulier’s dichotomy and his facts on local embeddability, Lee’s Dehn function, Johnson’s presentation, and Brown, Brin, and Hennig and Matucci for finite presentation are cited where they are used, as are the lemmas taken from the companion papers.
1.0.110.5281/zenodo.23065778
1.0.010.5281/zenodo.23050303

Paper, code and links

PDF

papers/configuration-lamps-v1.2.0.pdf, version 1.2.0, 1,321,935 bytes, 78 pages.

SHA-256 08f6116f2d089eced411b84932615492038e0b63760554ebd1813a94a0bbced0

Corrections

  1. Version 1.1.2. Credit added in the introduction: Gromov and Weiss; OpenAI’s announcement, Fournier-Facio, and Kun and Thom on non-sofic groups; Slofstra and Vidick, and Slofstra; Arzhantseva and Cherix; and work on channel simulation, memory costs and collision models.doi:10.5281/zenodo.23111927
  2. Version 1.1.1. Precision: a step in an appendix’s Choi-rank count is corrected (the bound is unchanged), and a set of coefficients is corrected from {±1, ±2−1/2} to {1, ±2−1/2}.doi:10.5281/zenodo.23108312
  3. Version 1.0.2. Credit made precise: Cornulier’s dichotomy and his facts on local embeddability, Lee’s Dehn function, Johnson’s presentation, and Brown, Brin, and Hennig and Matucci for finite presentation are cited where they are used, as are the lemmas taken from the companion papers.doi:10.5281/zenodo.23070871

All corrections, and what counts as one.

Cite

@misc{douglas2026configurationlamps,
  author        = {Douglas, Seth and Mghirbi, Nidhal},
  title         = {Amenable groups with nearly exponential sofic profile, and quantum channels that need nearly linear memory},
  year          = {2026},
  version       = {1.2.0},
  doi           = {10.5281/zenodo.23875973},
  url           = {https://doi.org/10.5281/zenodo.23875973},
  eprint        = {2610.04271},
  archiveprefix = {arXiv}
}

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