ZETAGROUPDYN_Zeta functions of groups and dynamical systems

群的Zeta函数通常是Dirichlet生成函数，编码关于无限群或有限群的无限族的算术信息，从而促进应用分析方法来研究数据。突出的例子分别是子群Zeta函数。反映有限指标子群分布的表示Zeta函数。给定群的有限维不可约表示。最近在理解算术群的表示Zeta函数方面取得了显著进展，例如数域上的整数环上的特殊线性群。Reidemister Nielsen不动点理论旨在研究连通紧流形上迭代自映射的不动点。自然产生的相关不变量，如尼尔森数和雷德迈斯特数，是通过适当的Zeta函数进行编码的。根据流形的基本群，这些Zeta函数可以看作是配备有自同态的群的动态Zeta函数。重要的结果包括幂零群上某些类型的自同态的Zeta函数和函数方程的合理性。我们的目标是将这两个领域连接起来，即。群的Zeta函数和动态Zeta函数，以获得基本的新见解。我们将以原创的方式在两个地区之间传递概念和方法，开辟两个地区之间的实质性联系。我们期待这将成为未来朝这一方向合作的跳板。

Zeta functions of groups are typically Dirichlet generating functions encoding arithmetic information about infinite groups - or infinite families of finite groups - and thereby facilitating the application of analytic methods to study the data. Prominent examples are subgroup zeta functions resp. representation zeta functions reflecting the distribution of finite index subgroups resp. finite dimensional irreducible representations of a given group. Significant advances were recently made in understanding representation zeta functions of arithmetic groups, such as the special linear group over the ring of integers of a number field.Reidemeister Nielsen fixed-point-theory is aimed at studying the fixed points of iterated self-maps on connected compact manifolds. Naturally arising relevant invariants, such as the Nielsen and Reidemeister numbers, are encoded by means of suitable zeta functions. In terms of the fundamental group of the manifold, these zeta functions can be seen as dynamical zeta functions of groups equipped with an endomorphism. Significant results include the rationality of such zeta functions and functional equations for certain types of endomorphisms on nilpotent groups.The proposed research is to be carried out via three PhD projects. Our aim is to connect the two areas, viz. zeta functions of groups and dynamical zeta functions, in order to obtain fundamental new insights. We will transfer concepts and methods between the two areas in an original way, pioneering substantial connections between the two areas. It is our expectation that this will form a springboard for future cooperations in this direction.