Groups of Intermediate Growth, an Introduction

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, Ahlfors, 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. We concentrate on study of the first construction, a finitely generated group G introduced by the first author to resolve Milnor’s question, and which became a prototype for further developments. Our Main Theorem shows that G has intermediate growth, i.e. superpolynomial and subexponential. Our proof is neither the shortest nor gives the best possible bounds. Instead, we attempt to simplify the presentation as much as possible by breaking the proof into a number of propositions of independent interest, supporting lemmas, and exercises. Along the way we prove two ‘bonus’ theorems: we show that G is periodic (every element has a finite order) and give a nearly linear time algorithm for the word problem in G. We hope that the beginner readers now have an easy time entering the field and absorbing what is usually viewed as unfriendly material. Let us warn the reader that this paper neither gives a survey nor presents a new proof of the Main Theorem. We refer to extensive survey articles [1, 2, 6] and a recent