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

This paper deals with the following topics. 1) Numerical invariants of countable groups (entropy, logarithmic volume, a nd drift); the fundamental inequality relating these invariants, and compar ison of generating sets on the basis of this inequality; Monte Carlo genera tion 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 gro ups. 5) Entropy in measure theory and in the theory of dynamical systems; new no tions of entropy of a decreasing sequence of measurable partitions and seco ndary entropy of K-automorphisms.