Spectral gaps of random finite-area hyperbolic surfaces

本项目的目标是研究Laplace-Beltrami算子谱的底部与有限区域非紧双曲曲面的谱的其余部分之间的谱间隙。所研究的随机曲面的一个模型是随机覆盖模型。固定一个双曲曲面，对于每个n，随机均匀地选取曲面的n次覆盖。该项目旨在确定是否n趋于无穷大，随机度n覆盖的固定表面具有几乎最佳的相对光谱间隙的概率趋于1。所讨论的谱隙必须是一个相对的，因为原固定曲面的特征值也是任何覆盖的特征值。最近Magee，Naud和Puder在紧致曲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，有限区域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，Selberg迹公式明显更复杂。在基本群中的字长和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第一个重点。这种方法的一个可能的结果是解决一个开放的问题，是否存在一个序列的连接双曲曲面的一般趋于无穷大，并具有最小的非-其光谱的零元素趋于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紧表面，都在这个项目的范围内。该项目可以采取的其他方向是研究非紧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。它也是感兴趣的双曲曲面产生的“光谱间隙”以外的其他在0;有限区域的非紧双曲曲面的唯一真正的光谱间隙可以是0和1/4之间。另一方面，这些表面有一个有趣的嵌入特征值的现象，可以研究随机覆盖。

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.