Voskou, Marios;
(2025)
Distributional Results for Geodesic Segments in the Hyperbolic Plane.
Doctoral thesis (Ph.D), UCL (University College London).
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Abstract
For Γ a cofinite Fuchsian group, and l a fixed closed geodesic, we study refined asymptotics of the number of images of l over Γ that have a distance from l less than or equal to X. In particular, we partition the images into four cases, according to orientation, and prove that they all contribute asymptotically one-fourth of the total. This was originally studied by A. Good. For Γ_1 the stabilizer of l, this is equivalent to counting the number of double cosets in Γ_1\Γ/Γ_1 with prescribed signs for its entries, according to a certain growth parameter. We achieve this by developing new modified relative trace formulae, as well as bounds for hyperbolic periods in mean square. We give a new concrete proof of the error bound O(X^(2/3)) that appears in the works of Good and Hejhal. Furthermore, we prove a new bound O(X^(1/2) logX) for the mean square of the error. To that end, we obtain large sieve inequalities with weights the hyperbolic periods of Maaß forms of even weight. This is inspired by work of Chamizo, who proved a large sieve inequality with weights the values of Maaß forms of weight 0. We also prove Ω results, supporting the conjectural best error term O_ε(X^(1/2+ε))$. For particular arithmetic groups, we provide interpretations in terms of correlation sums of the number of ideals of norm at most X in associated number fields, generalizing previous examples due to Hejhal.
| Type: | Thesis (Doctoral) |
|---|---|
| Qualification: | Ph.D |
| Title: | Distributional Results for Geodesic Segments in the Hyperbolic Plane |
| Open access status: | An open access version is available from UCL Discovery |
| Language: | English |
| Additional information: | Copyright © The Author 2025. Original content in this thesis is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) Licence (https://creativecommons.org/licenses/by-nc/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10203381 |
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