Groups and Topological Dynamics

这个项目探索了数学的两个分支之间的新发现的联系：动力系统理论和群论。前者研究复杂和混沌系统的行为，而后者通常研究（例如几何）结构的对称性群。很明显，有趣的对称群与许多混沌动力系统自然地联系在一起。这种联系使得回答群论和动力系统理论中一些长期存在的开放性问题成为可能。例如，证明了低复杂性的动力系统可以用来构造亚指数增长的简单群，这是一个30年来一直悬而未决的问题。项目负责人计划利用混沌动力系统理论的方法解决群论中的问题，并利用群论更好地理解动力系统。本项目将探索的方向是研究动力系统和群的不同复杂性度量之间的关系：增长率、熵、模式复杂性、“有限型”条件等。本项目将研究与动力系统自然相关的群：动力系统和虚群的拓扑满群、拓扑空间自覆盖的迭代单群、层合的完整群等。从“经典”群论的角度来看，它们通常是外来的，并且是有趣例子的良好来源，特别是从群的渐近性质（例如适应性和增长）的角度来看。新的拓扑和动力学方法已被证明是研究这类群的有效工具。例如，几乎所有目前已知的非初等可服从群的例子都是用拓扑动力系统来构造和研究的。本研究项目旨在利用拓扑动力学和典型群类群对若干群论问题有新的认识，并对一些长期存在的开放性问题（如可能的中间词缓慢增长、群的可适应性、扭转群的新例子、构造具有奇异性质的有限呈现群、二次多项式迭代一元群的增长研究等）进行研究。PI还预计群论将有效地研究动力系统中的问题。

This project explores newly discovered connections between two branches of Mathematics: the theory of dynamical systems and group theory. The former studies behavior of complex and chaotic systems, while the latter usually studies groups of symmetries of (for example geometric) structures. It has become clear that interesting groups of symmetries are naturally associated with many chaotic dynamical systems. This connection makes it possible to answer some longstanding open questions in group theory, as well as in the theory of dynamical systems. For example, it was shown that dynamical systems of low complexity can be used to construct simple groups of sub-exponential growth, the existence of which was an open question for 30 years. The principal investigator plans to use the methods of the theory of chaotic dynamical systems to solve problems in group theory and to use group theory to better understand dynamical systems. A direction which will be explored in this project is the study of relations between different measures of complexity of dynamical systems and groups: rates of growth, entropy, pattern complexity, "finite type" conditions, etc.This project will study groups naturally associated with dynamical systems: topological full groups of dynamical systems and etale groupoids, iterated monodromy groups of self-coverings of topological spaces, holonomy groups of laminations, etc. They are often exotic from the point of view of "classical" group theory and are good sources of interesting examples, especially from the point of view of asymptotic properties of groups (for example amenability and growth). New topological and dynamical methods have proven to be effective tools in the study of such groups. For instance, almost all currently known examples of non-elementary amenable groups are constructed and studied using topological dynamical systems. This research program aims to shed new light on several group theoretic questions using topological dynamics and etale groupoids, and to work on some long-standing open questions (for example, on possible slow intermediate word growth, amenability of groups, new examples of torsion groups, constructing finitely presented groups with exotic properties, studying growth of iterated monodromy groups of quadratic polynomials, etc.). The PI also anticipate that group theory will be effective in studying problems in dynamical systems.