Bibliographic record
Abstract
We describe a procedure called panel collapse for replacing a CAT(0) cube complex \Psi by a "lower complexity" CAT(0) cube complex \Psi_\bullet whenever \Psi contains a codimension-2 hyperplane that is extremal in one of the codimension-1 hyperplanes containing it. Although \Psi_\bullet is not in general a subcomplex of \Psi , it is a subspace consisting of a subcomplex together with some cubes that sit inside \Psi "diagonally." The hyperplanes of \Psi_\bullet extend to hyperplanes of \Psi . Applying this procedure, we prove: if a group G acts cocompactly on a CAT(0) cube complex \Psi , then there is a CAT(0) cube complex \Omega so that G acts cocompactly on \Omega and for each hyperplane H of \Omega , the stabiliser in G of H acts on H essentially. Using panel collapse, we obtain a new proof of Stallings's theorem on groups with more than one end. As another illustrative example, we show that panel collapse applies to the exotic cubulations of free groups constructed by Wise in [44]. Next, we show that the CAT(0) cube complexes constructed by Cashen and Macura in [7] can be collapsed to trees while preserving all of the necessary group actions. (It also illustrates that our result applies to actions of some non-discrete groups.) We also discuss possible applications to quasi-isometric rigidity for certain classes of graphs of free groups with cyclic edge groups. Panel collapse is also used in forthcoming work of the first-named author and Wilton to study fixed-point sets of finite subgroups of Out (F_n) on the free splitting complex. Finally, we apply panel collapse to a conjecture of Kropholler, obtaining a short proof under a natural extra hypothesis.
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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 | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.002 | 0.007 |
| Research integrity | 0.001 | 0.004 |
| Insufficient payload (model declined to judge) | 0.020 | 0.005 |
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".