# Mil nor's problem on the growth of groups and its consequences

# Mil nor's problem on the growth of groups and its consequences

This chapter presents a survey of results related to John Milnor's problem on group growth. The notion of group growth first appeared in 1955 in a paper of A. S. Schwarz, but it remained virtually unnoticed for over a decade. The situation changed after Milnor's papers from 1968, which sparked significant interest in this area. Particularly influential were two problems raised in these papers: the characterization of groups of polynomial growth and the question of the existence of groups of intermediate growth. The chapter discusses the cases of polynomial growth and exponential but not uniformly exponential growth; the main part of this chapter is devoted to the intermediate (between polynomial and exponential) growth case. A number of related topics (growth of manifolds, amenability, asymptotic behavior of random walks) are considered, and a number of open problems are suggested.

*Keywords:*
group growth, polynomial growth, intermediate growth, exponential growth, manifold growth, amenability, asymptotic behavior, random walks

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