Cogrowth series for free products of finite groups
Bibliographic record
Abstract
Given a finitely generated group with generating set [Formula: see text], we study the cogrowth sequence, which is the number of words of length [Formula: see text] over the alphabet [Formula: see text] that are equal to the identity in the group. This is related to the probability of return for walks on the corresponding Cayley graph. Muller and Schupp proved the generating function of the sequence is algebraic when [Formula: see text] has a finite-index-free subgroup (using a result of Dunwoody). In this work, we make this result effective for free products of finite groups: we determine bounds for the degree and height of the minimal polynomial of the generating function, and determine the minimal polynomial explicitly for some families of free products. Using these results we are able to prove that a gap theorem holds: if [Formula: see text] is a finite symmetric generating set for a group [Formula: see text] and if [Formula: see text] denotes the number of words of length [Formula: see text] over the alphabet [Formula: see text] that are equal to [Formula: see text] then [Formula: see text] is either [Formula: see text], [Formula: see text] or at least [Formula: see text].
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 teacher head, 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".