Random Sampling of Trivial Words in Finitely Presented Groups
Bibliographic record
Abstract
We describe a Metropolis Monte Carlo algorithm for random sampling of freely reduced words equal to the identity in a finitely presented group. The algorithm samples from a stretched Boltzmann distribution π(w)=(|w|+1)αβ|w|·Z-1where |w| is the length of a word w, α and β are parameters of the algorithm, and Z is a normalizing constant. It follows that words of the same length are sampled with the same probability. The distribution can be expressed in terms of the cogrowth series of the group, which allows us to relate statistical properties of words sampled by the algorithm to the cogrowth of the group, and hence its amenability. We have implemented the algorithm and applied it to several group presentations including the Baumslag–Solitar groups, some free products studied by Kouksov, a finitely presented amenable group that is not subexponentially amenable (based on the basilica group), the genus 2 surface group, and Richard Thompson’s group F.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.010 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".