Groups of intermediate growth

有限生成群的增长是一个重要的概念。它允许人们大规模地测量和比较群体，并在几何、拓扑、分析、概率、动力学和其他数学领域有许多应用。群的增长可以是多项式增长、指数增长或介于多项式和指数之间。中间生长的类群是神秘的。米尔诺关于这个班级是否空的问题已经提出了超过15年。1983年，PI构造了无数具有不同增长类型的中间增长群。这导致了第一个不可数的2生成群的拟等距类的构造，并给出了标记群空间中康托尔子集的第一个显式构造。尽管在中间生长群的研究中取得了这些成功，但仍有许多基本的开放性问题。主要的开放问题包括：关于有限表示的中间增长群的存在性问题，以及关于多项式增长与中间增长之间“差距”大小的问题（Gap猜想）。在其他重要问题中，关于遗传的正无穷群和中间生长的简单群的存在性的问题。本建议解决这些问题和其他相关问题。PI将Gap猜想约简为正无穷群，其中考虑了上述两类和分支群的类。技术包括在有根树上的群体行动和动力系统的方法。有限生成群的增长与随机漫步理论、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的影响。基于中间生长的自相似群构建的Schreier图和分形可能与我们理解生物学、化学和人口统计学研究中的一些过程有关。对中间生长群体的研究产生的算法不同于以往的算法，在科学和技术上有应用。在数学中，增长的概念不仅对几何群论，而且对算子代数、拓扑学、几何、动力系统、泛函分析、概率论、离散数学、微分方程等领域都具有重要意义。成长的主题非常适合研究生甚至本科课程，因为它涉及到现代数学中的许多相关主题。PI将通过同行评议出版物以及向国内外各类听众举办研讨会和专题讨论会、特邀讲座和演讲等方式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.