Exceptional eigenvalue density for thin groups

Christopher Lutsko

arXiv:2608.18236v2 Primary: math.SP Secondary: math.GT

Abstract

We prove a limit multiplicity conjecture of Hee Oh for principal congruence covers of geometrically finite hyperbolic manifolds, with an explicit power-saving rate. The rate is governed by the return of Patterson--Sullivan shadows through a fixed compact core, giving a geometric interpretation to the exceptional eigenvalue density. For convex-cocompact groups, a packing argument for enlarged shadows gives a stronger rate.

Disclosure

“= 2 exp − ℓq . δΓ − ρ Since ℓq = log q − O(1), taking ϑ < 1/2 sufficiently close to 1/2 proves (1.2). Acknowledgements The author used OpenAI’s ChatGPT for research brainstorming, literature discovery, checking intermediate arguments and calculations, and editorial revision. ChatGPT was not treated as an author or as an independent source of mathematical authority. The author assumes full”
PDF page 10

Classification: Brainstorming or outlining
Multiplier: 2
Verified: 2026-08-22T02:26:08.472129Z

Structural counts

Pages 11 pdf

Theorems 3 source

Lemmas 3 source

Propositions 0 source

Corollaries 0 source

Definitions 0 source

Displayed equations 57 source

Bibliography entries 16 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file PS_shadow_density.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.