Clearly, once we have constructed one group of intermediate growth, we can find many others, e.g. by taking direct products of our group either with itself or with groups of polynomial growth, or by taking finite extensions, etc. Many other constructions of groups of intermediate growth were offered as well. In this chapter we first describe a generalization, due to Grigorchuk himself, of the construction of the previous chapter. Then we will describe other approaches to the same groups; these approaches lead to many other interesting groups. Of these we supply some examples, but no proofs.

The General Grigorchuk Groups

We consider again transformations of the unit interval with the dyadic rationals removed. We divide that interval in the same way, and let E and P denote the same transformations, as before. We now let Γ be the group generated by four transformations a, b, c, d, where a = P is the same as before, and each of b, c, d acts on the subintervals (0, 1/2), (1/2, 3/4), … by some sequence of transformations P and E, e.g. P, P, P, E, E, P, E, E, E, …. Each such sequence is allowed, but we assume that the three sequences defining b, c, d are related by requiring that on each subinterval two of b, c, d act as P, and the third one as E.