Groups of intermediate growth

生成群的增长是一个重要的概念。它允许一个测量和比较群体在大规模上，并有许多应用在几何，拓扑，分析，概率，动力学和其他领域的数学。群的增长可以是多项式的、指数的或介于多项式和指数之间的。中等增长的群体类别是神秘的。米尔诺关于这个班是否空无一人的问题已经公开了15年多。在1983年，PI构建了无数个具有不同增长类型的中间增长组。这导致了第一次建设的不可数许多准等距类的2生成的群体，并给予了第一个明确的建设康托子集的空间中的标记群。尽管在对中等增长群体的研究中取得了这些成功，但仍然存在许多根本性的问题。主要的开放问题包括：关于中间增长的非线性群的存在性问题，以及关于多项式增长和中间增长之间的“间隙”大小的问题（间隙猜想）。在其他重要的问题是问题的存在遗传公正无限群和简单的群体的中间增长。本提案涉及这些问题和其他相关问题。PI将差距猜想约化为恰好无限群，其中包括对上述两类和分支群类的考虑。技术包括根树上的群作用和动力系统的方法。生成群的增长与随机行走理论，分形几何，晶体和准晶体，编码理论，形式语言，有限自动机和细胞自动机的动力学，通信网络建模，Kolmogorov复杂性和许多其他主题有关。作为当前研究的一部分，所获得的结果对这些领域以及对通信网络、密码学和运输系统的科学和技术理解具有潜在影响。Schreier图和分形的基础上构建的自相似组的中间增长可能是相关的，我们的理解在生物学，化学和人口研究中的一些过程。从研究中间增长群中产生的算法与以前使用的算法不同，并且在科学和技术中有应用。 在数学中，增长的概念不仅对几何群论很重要，而且对算子代数，拓扑学，几何学，动力系统，泛函分析，概率论，离散数学，微分方程和其他领域也很重要。增长的主题非常适合研究生甚至本科课程，因为它涉及现代数学中的许多相关主题。PI将通过同行评审的出版物和举办研讨会和学术讨论会，邀请国内和国际各类受众进行讲座和演讲来传播这项研究的结果。

Growth of finitely generated groups is an important notion. It allows one to measure and compare groups on a large scale and has numerous applications in geometry, topology, analysis, probability, dynamics and other areas of mathematics. The growth of a group can be polynomial, exponential or intermediate between polynomial and exponential. The class of groups of intermediate growth is mysterious. Milnor's question as to whether this class is empty was open for more than 15 years. In 1983 the PI constructed uncountably many groups of intermediate growth with different types of growth. This led to the first construction of uncountably many quasi-isometry classes of 2-generated groups and gave the first explicit construction of a Cantor subset in the space of marked groups. Despite these successes in the study of groups of intermediate growth, there are still many fundamental open problems. The main open problems include: the question about the existence of finitely presented groups of intermediate growth, and the question (Gap Conjecture) about the size of the "gap" between polynomial growth and intermediate growth. Among other important problems are the questions about the existence of hereditary just-infinite groups and of simple groups of intermediate growth. This proposal addresses these and other related questions. The PI has a reduction of the Gap Conjecture to just-infinite groups, which includes the consideration of the above two classes and the class of branch groups. Techniques include group actions on rooted trees and the methods of dynamical systems.Growth of finitely generated groups is related with the theory of random walks, the geometry of fractals, crystals and quasi-crystals, coding theory, formal languages, dynamics of finite automata and cellular automata, modeling of communication networks, Kolmogorov complexity and many other topics. Results obtained as a part of the current research have potential implications for these areas, and for the scientific and technological understanding of communication networks, cryptography, and transportation systems. Schreier graphs and fractals constructed on the basis of self-similar groups of intermediate growth may be relevant for our understanding of some processes studied in biology, chemistry and demographic studies. Algorithms arising from the study of groups of intermediate growth are different from those used before, and have applications in science and technology. In mathematics, the notion of growth is important not only for Geometric Group Theory, but also for Operator Algebras, Topology, Geometry, Dynamical Systems, Functional Analysis, Probability Theory, Discrete Mathematics, Differential Equations and other fields. The subject of growth is very suitable for graduate and even undergraduate courses, since it touches on many relevant topics in modern mathematics. The PI will disseminate the results of this research through peer reviewed publication and by giving seminar and colloquium talks, invited lectures and presentations to various types of audiences both domestically and internationally.