Dynamic theory of growth in groups: Entropy, boundaries, examples

Abstract

This paper deals with the following topics. 1) Numerical invariants of countable groups (entropy, logarithmic volume, and drift); the fundamental inequality relating these invariants, and comparison of generating sets on the basis of this inequality; Monte Carlo generation of groups. 2) An ergodic method for constructing and studying the boundaries of random walks, the entropy of the boundary polymorphism, and their relationship to the fundamental inequality. 3) A geometric realization of free soluble groups, their boundaries, and a geometric approach to the construction of normal forms in groups. 4) Local and locally free groups and calculation of constants for these groups. 5) Entropy in measure theory and in the theory of dynamical systems; new notions of entropy of a decreasing sequence of measurable partitions and secondary entropy of K-automorphisms.