Bowditch Taut Spectrum and Dimensions of Groups
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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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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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 it