- Experiments with Pants Covers of the Modular Surface
- Project Year:
2020
- REU Student (s):
Colin Fan | Rutgers University-New Brunswick NJ
| Saket Shah | Princeton University NJ
- Student 1 Institution:
Rutgers University-New Brunswick
- Student 2 Institution:
Princeton University
- Project Mentor:
Alex Kontorovich
- Project Mentor Area:
Mathematics
- Project Abstract:
Let X and Y be two finite-area hyperbolic Riemann surfaces. Since X and Y are both hyperbolic, we know that they must both be uniformized by ℍ ; however, the maps ℍ → X and ℍ → Y are not in general of finite degree. In general, guaranteeing that X and Y share conformally equivalent finite covers is too much to hope. The Ehrenpreis conjecture states that we can achieve something close: in particular, for any K>1, we can find finite covers X' → X and Y' → Y such that X' and Y' are K-quasiconformally equivalent; in other words, we can always find finite covers which have little distortion of angles relative to each other. The conjecture was recently settled in the case of closed surfaces in 2015 by Kahn and Markovic, but there has been no such resolution for punctured Riemann surfaces, and in fact the truth of the conjecture remains contested. The modular surface is a model example of a punctured Riemann surface of finite area. In this project, we numerically find and study finite covers of the modular surface constructed by gluing together immersed hyperbolic pants and "degenerate pants", in which we allow one cuff to degenerate into a cusp, subject to similar technical constraints. We hope that the work will help in generating insight towards an angle of attack in a proof of the cusped Ehrenpreis conjecture inspired by the methods used in the proof of the case of closed surfaces.
