Bi-invariant metrics and quasi-morphisms on groups of Hamiltonian diffeomorphisms of surfaces
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Bibliographic record
Abstract
Let Σ g be a closed orientable surface of genus g and let Diff 0 (Σ g , area ) be the identity component of the group of area-preserving diffeomorphisms of Σ g . In this paper, we present the extension of Gambaudo–Ghys construction to the case of a closed hyperbolic surface Σ g , i.e. we show that every nontrivial homogeneous quasi-morphism on the braid group on n strings of Σ g defines a nontrivial homogeneous quasi-morphism on the group Diff 0 (Σ g , area ). As a consequence we give another proof of the fact that the space of homogeneous quasi-morphisms on Diff 0 (Σ g , area ) is infinite-dimensional. Let Ham (Σ g ) be the group of Hamiltonian diffeomorphisms of Σ g . As an application of the above construction we construct two injective homomorphisms Z m → Ham (Σ g ), which are bi-Lipschitz with respect to the word metric on Z m and the autonomous and fragmentation metrics on Ham (Σ g ). In addition, we construct a new infinite family of Calabi quasi-morphisms on Ham (Σ g ).
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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 | 0.002 | 0.005 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.002 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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