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.
