Cubulated groups: thickness, relative hyperbolicity, and simplicial boundaries
Bibliographic record
Abstract
Let G be a group acting geometrically on a CAT(0) cube complex \mathbf X . We prove first that G is hyperbolic relative to the collection \mathbb P of subgroups if and only if the simplicial boundary \partial_{\triangle} \mathbf X is the disjoint union of a nonempty discrete set, together with a pairwise-disjoint collection of subcomplexes corresponding, in the appropriate sense, to elements of \mathbb P . As a special case of this result is a new proof, in the cubical case, of a Theorem of Hruska and Kleiner regarding Tits boundaries of relatively hyperbolic CAT(0) spaces. Second, we relate the existence of cut-points in asymptotic cones of a cube complex \mathbf X to boundedness of the 1-skeleton of \partial_{\triangle} \mathbf X . We deduce characterizations of thickness and strong algebraic thickness of a group G acting properly and cocompactly on the CAT(0) cube complex \mathbf X in terms of the structure of, and nature of the G -action on, \partial_{\triangle} \mathbf X . Finally, we construct, for each n\geq 0, k\geq 2 , infinitely many quasi-isometry types of group G such that G is strongly algebraically thick of order n , has polynomial divergence of order n+1 , and acts properly and cocompactly on a k -dimensional CAT(0) cube complex.
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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 | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".