Spectral gaps of random finite-area hyperbolic surfaces

随机有限面积双曲曲面的谱隙

• 批准号: 2457694
• 金额: --
• 依托单位: Durham University
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2457694
• 关键词: Spectral; gaps; random; finite; area

The goal of this project is to investigate the spectral gap between the bottom of the spectrum of the Laplace-Beltrami operator and the rest of the spectrum for finite-area, non-compact hyperbolic surfaces.One model of random surfaces studied will be the random covering model. One fixes a hyperbolic surface, and for each n, picks a degree n cover of the surface uniformly at random. The project seeks to determine whether as n tends to infinity, the probability that a random degree n cover of a fixed surface has an almost optimal relative spectral gap tends to 1. The spectral gap discussed must be a relative one, since the eigenvalues of the original fixed surface are also eigenvalues of any cover.Recent progress on this problem for compact surfaces has been made by Magee, Naud, and Puder. The rough plan is to follow the approach of (ibid.), however, the finite-area non-compact setting poses two new significant challenges:1. The Selberg trace formula is significantly more involved due to contributions from the continuous spectrum.2. There is no uniform comparison between word length in the fundamental group and the corresponding geodesic length in the surface, due to the presence of cusps in the surface.Overcoming these challenges will be a first focus of the project.One possible consequence of this approach is to resolve an open problem whether there exist a sequence of connected hyperbolic surfaces with genera tending to infinity and with smallest non-zero element of their spectrum tending to 1/4. Any alternative approach to this problem, particularly for finite-area non-compact surfaces, is within the scope of this project.Other directions that the project can take is the study of alternative random models of non-compact surfaces, such as Weil-Petersson model and/or combinatorial model introduced by Brooks-Makover. It is also of interest to produce hyperbolic surfaces with `spectral gaps' other than at 0; for finite-area non-compact hyperbolic surfaces the only true spectral gaps can be between 0 and 1/4. On the other hand, these surfaces have an interesting phenomenon of embedded eigenvalues that can be studied for random covers.

这个项目的目标是研究有限面积、非紧双曲曲面的拉普拉斯-贝尔特拉米算子的光谱底部与光谱其余部分之间的光谱间隙。随机表面的一个模型是随机覆盖模型。一种是固定一个双曲曲面，对于每一个n，均匀随机地选择一个n度的曲面。该项目旨在确定当n趋于无穷大时，固定表面的随机度n覆盖具有几乎最优相对光谱间隙的概率是否趋于1。所讨论的光谱间隙必须是相对的，因为原始固定表面的特征值也是任何覆盖物的特征值。Magee、Naud和Puder最近在紧致表面的这个问题上取得了进展。粗略的计划是遵循（同上）的方法，然而，有限面积的非紧凑设置提出了两个新的重大挑战：1。由于连续谱的贡献，塞尔伯格示踪公式明显更加复杂。由于曲面中存在尖点，基本群中的单词长度与曲面中相应的测地线长度之间没有统一的比较。克服这些挑战将是该项目的首要重点。这种方法的一个可能结果是解决了一个开放的问题，即是否存在一串连通的双曲曲面，其属趋于无穷，其谱的最小非零元趋于1/4。任何解决这个问题的替代方法，特别是对于有限面积的非致密表面，都在本项目的范围内。该项目可以采取的其他方向是研究非紧凑表面的替代随机模型，例如Weil-Petersson模型和/或Brooks-Makover引入的组合模型。产生除0以外具有“谱隙”的双曲曲面也是有意义的；对于有限面积的非紧双曲曲面，真正的谱隙只能在0到1/4之间。另一方面，这些表面有一个有趣的嵌入特征值现象，可以研究随机覆盖。