Milnor's Problem on the Growth of Groups and its Consequences

TL;DR

This survey explores Milnor's problem on group growth, focusing on intermediate growth cases and related topics like manifolds, amenability, and random walks, highlighting open problems and recent results.

Contribution

It provides a comprehensive overview of Milnor's problem, emphasizing the intermediate growth case and connecting it with related mathematical areas.

Findings

Discussion of polynomial and exponential growth cases

Analysis of intermediate growth and open problems

Connections to manifolds, amenability, and random walks

Abstract

We present a survey of results related to the Milnor's problem on group growth. We discuss the cases of polynomial growth, exponential but not uniformly exponential growth, but the main part of the article 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) is considered, and a number of open problems is suggested.