Large degree covers and sharp resonances of hyperbolic surfaces
Bibliographic record
Abstract
Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> be a convex co-compact discrete group of isometries of the hyperbolic plane <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℍ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> , and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>=</mml:mo> <mml:mi>Γ</mml:mi> <mml:mo>∖</mml:mo> </mml:mrow> <mml:msup> <mml:mi>ℍ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> the associated surface. In this paper we investigate the behaviour of resonances of the Laplacian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Δ</mml:mi> <mml:mover accent="true"> <mml:mi>X</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> </mml:msub> </mml:math> for large degree covers of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> given by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover accent="true"> <mml:mi>X</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> <mml:mo>=</mml:mo> <mml:mover accent="true"> <mml:mi>Γ</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> <mml:mrow> <mml:mo>∖</mml:mo> </mml:mrow> <mml:msup> <mml:mi>ℍ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover accent="true"> <mml:mi>Γ</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> <mml:mo>⊲</mml:mo> <mml:mi>Γ</mml:mi> </mml:mrow> </mml:math> is a finite index normal subgroup of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> . Using techniques of thermodynamical formalism and representation theory, we prove two new existence results of sharp non-trivial resonances close to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>Re</mml:mi> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>δ</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> , in the large degree limit, for abelian covers and infinite index congruence subgroups of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>SL</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ℤ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> .
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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".