Geometrical compactifications of geodesic flows and path structures
Bibliographic record
Abstract
In this paper, we construct a geometrical compactification of the geodesic flow of non-compact complete hyperbolic surfaces $Σ$ without cusps having finitely generated fundamental group. We study the dynamical properties of the compactified flow, for which we show the existence of attractive circles at infinity. The geometric structure of ${\mathrm{T}}^1Σ$ for which this compactification is realized is the pair of one-dimensional distributions tangent to the stable and unstable horocyles of ${\mathrm{T}}^1Σ$. This is a Kleinian path structure, that is a quotient of an open subset of the flag space by a discrete subgroup $Γ$ of ${\mathrm{PGL}}_3(\mathbb{R})$. Our study relies on a detailed description of the dynamics of ${\mathrm{PGL}}_3(\mathbb{R})$ on the flag space, and on the construction of an explicit fundamental domain for the action of $Γ$ on its maximal open subset of discontinuity in the flag space.
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How this classification was reachedexpand
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 | 0.003 | 0.006 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".