Groups of Intermediate Growth: an Introduction for Beginners

The study of growth of groups has a long and remarkable history spanning over much of the twentieth century, and goes back to Hilbert, Poincare, Alphors, etc. In 1968 it became apparent that all known classes of groups have either polynomial or exponential growth, and John Milnor formally asked whether groups of intermediate growth exist. The first such examples were introduced by the first author two decades ago [4] (see also [3, 5]), and since then there has been an explosion in the number of works on the subject. While new techniques and application have been developed, much of the literature remains rather specialized, accessible only to specialists in the area. This paper is an attempt to present the material in an introductory manner, to the reader familiar with only basic algebraic concepts.