Houghton’s group has superpolynomial sofic profile
In plain wordsA finitely presented amenable group that moves points along three rays has a finite piece whose imitation by permutations needs more than polynomially many points, which answers a published question; a published extension theorem that would give a quadratic bound fails as stated.
The result
As the paper states it, in briefHoughton’s finitely presented elementary amenable group has a finite chunk whose sofic profile is superpolynomial.
Questionson the index
- Answered Can a sofic group have a sofic profile that is unbounded and not linear? posed by others
- Answered Is the sofic profile of Houghton’s group polynomial?
How this was checked
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Novelty and credit
- Novelty
- New, with named precedent: Slofstra’s finitely presented group has a chunk with a larger exponent but is not known to be sofic; the paper knows of no earlier amenable group with superpolynomial sofic profile. It answers the last question of Cornulier’s Problem 3.18 and locates an error in his extension theorem.
Versionsfive, each with its own DOI
| Version | Date | DOI | Notes |
|---|---|---|---|
| 1.2.0 | 10.5281/zenodo.23875978 | Adds a sweep form of the counting criterion: a quadratic Dehn function for H3, or a quadratic bound for block coherence, would give a lower exponent of 1/2, matching the upper exponent up to a logarithm. The unconditional bounds are unchanged, and the open problem is sharpened. PDF hosted here. | |
| 1.1.2 | 10.5281/zenodo.23070869 | Correction: Credit made precise: Cornulier’s linear lower bound and bounded cases, Lee’s reduction of the Dehn function, Elek and Szabó’s construction, Arzhantseva and Cherix, Röver, and the known birational consequences are credited where they are used; Slofstra’s earlier chunk is described exactly, with its exponent, near 1/4, larger than ours. The mathematics is unchanged. | |
| 1.1.1 | 10.5281/zenodo.23065750 | ||
| 1.1.0 | 10.5281/zenodo.23048875 | ||
| 1.0.0 | 10.5281/zenodo.23027625 |
Paper, code and links
papers/houghton-v1.2.0.pdf, version 1.2.0, 728,797 bytes, 37 pages.SHA-256 4d337e8d952b63ef39038c6432a0ef08ccfb58856486a3ee8d396f94f8cea2fb
- DOI
10.5281/zenodo.23875978; all versions 10.5281/zenodo.23027624
Corrections
- Version 1.1.2. Credit made precise: Cornulier’s linear lower bound and bounded cases, Lee’s reduction of the Dehn function, Elek and Szabó’s construction, Arzhantseva and Cherix, Röver, and the known birational consequences are credited where they are used; Slofstra’s earlier chunk is described exactly, with its exponent, near 1/4, larger than ours. The mathematics is unchanged.doi:10.5281/zenodo.23070869
All corrections, and what counts as one.
Cite
@misc{douglas2026houghton,
author = {Douglas, Seth and Mghirbi, Nidhal},
title = {Houghton's group {H3} has superpolynomial sofic profile},
year = {2026},
version = {1.2.0},
doi = {10.5281/zenodo.23875978},
url = {https://doi.org/10.5281/zenodo.23875978}
}