Analytic and Reidemeister torsion for non-compact locally symmetric spaces

数论研究领域的一个新进展是基于算术群的上同调挠应对应于有限域上的伽罗瓦表示的认识。由于这样的表示是数论中的核心兴趣，因此关于上同调挠的存在性的问题是极其重要的。对于上紧算术群，在几种情形下得到了上同调挠的渐近性的结果.然而，许多算术群不是余紧的。这是这种情况下，即使是那些群体出现的最自然的，这是最有趣的角度伽罗瓦表示，例如主同余子群的整数。对于这样的群，关于上同调挠的存在性和大小的问题通常是公开的。我的研究项目的主要目标是表明，也为算术群，这不是cocompact的上同调扭转指数增长。作为一个精确的定量陈述，我想确定这个渐近增长中的主导项与相应的L2挠率，其渐近行为是已知的。为了实现这些目标，我想研究有限体积的非紧局部对称空间的解析挠，特别是它与Reidemeister挠的关系以及它在高秩情况下的一些基本性质，我的研究的主要方法将基于几何分析技术在非紧和奇异流形上的应用，这是由斯坦福大学的Rafe Mazzeo教授特别开发的。由于这些技术是为非常一般的情况下开发的，他们在我看来也适用于各种变形的几何和分析结构，我想在我的证明在几个地方。此外，我打算使用的方法，从几何分析的情况下，局部对称空间的应用方法也作为基础，为进一步的研究项目，这不是本申请的内容。我想特别提到我的目标是研究连续频谱的拉普拉斯运营商使用的方法几何散射理论。

A new development in the research area of Number Theory is based on the insight that cohomological torsion of arithmetic groups should correspond to Galois representations over finite fields. Since such representations are of central interest in Number Theory, the question about the existence of cohomological torsion is therefore extremely important. This question is usually meant in an asymptotic sense.For cocompact arithmetic groups, results about the asymptotic behaviour of cohomological torsion were obtained in several situations. However, a lot of arithmetic groups are not cocompact. This is the case even for those groups which arise most naturally and which are most interesting from the point of view of Galois representations, for example principal congruence subgroups over the integers. For such groups the question about the existence and size of cohomological torsion is in general open. The main goal of my research project is to show that also for arithmetic groups which are not cocompact the cohomological torsion grows exponentially. As a precise quantitative statement I want to identify the leading term in this asymptotic growth with the corresponding L2 torsion, whose asymptotic behaviour is already known. In order to achieve these goals I want to investigate the analytic torsion of non-compact locally symmetric spaces of finite volume, in particular its relation to Reidemeister torsion as well as some of its basic properties in the higher rank situation.The main method of my research will be based on the application of techniques from Geometric Analysis on non compact and singular manifolds, which were in particular developed by Professor Rafe Mazzeo from Stanford. Since these techniques were developed for very general situations, they are on my opinion also applicable to the various deformations of the geometric and analytic structure that I want to use in my proof at several places.Furthermore, I intend to use the approach of applying methods from Geometric Analysis to the case of locally symmetric spaces also as a basis for further research projects which are not the content of this application. I would like to mention in particular my goal to study the continuous spectrum of the Laplace operator using methods of geometric scattering theory.