Bowditch Taut Spectrum and Dimensions of Groups
Bibliographic record
Abstract
For a finitely generated group G, let H(G) denote Bowditch’s taut loop length spectrum. We prove that if G=(A∗B)/⟨⟨R⟩⟩ is a C′(1/12) small cancellation quotient of a the free product of finitely generated groups, then H(G) is equivalent to H(A)∪H(B). We use this result together with bounds for cohomological and geometric dimensions, as well as Bowditch’s construction of continuously many non-quasi-isometric C′(1/6) small cancellation 2-generated groups to obtain our main result: Let G denote the class of finitely generated groups. The following subclasses contain continuously many one-ended non-quasi-isometric groups: (1) {G∈G:cd_(G)=2andgd_(G)=3}; (2) {G∈G:cd__(G)=2andgd__(G)=3}; (3) {G∈G: cdQ(G)=2andcdZ(G)=3}. On our way to proving the aforementioned results, we show that the classes defined above are closed under taking relatively finitely presented C′(1/12) small cancellation quotients of free products; in particular, this produces new examples of groups exhibiting an Eilenberg–Ganea phenomenon for families. We also show that if there is a finitely presented counterexample to the Eilenberg–Ganea conjecture, then there are continuously many finitely generated one-ended non-quasi-isometric counterexamples.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.000 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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 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".