Soluble Groups of Polynomial Growth

As already mentioned, one of the major results of our subject is M. Gromov's converse of Theorem 4.1: a group of polynomial growth is nilpotentby-finite [Gro 81]. This was proved first by J. Milnor [Mi 68] and J. Wolf [Wo 68] for soluble groups, and that result is our next goal. We start with:

Theorem 5.1Let G be a finitely generated group of subexponential growth. Then the commutator subgroup G′ of G is also finitely generated.

Corollary 5.2A finitely generated soluble group of subexponential growth is polycyclic.

Proof Let G be soluble of polynomial growth. The preceding theorem and induction imply that all commutator subgroups of G are finitely generated, and then so are the abelian factor groups G(i)/G(i+1). Thus these factor groups are polycyclic, and since only finitely many of them are non-trivial, so is G. QED

Proof of Theorem 5.1. Since G/G′ is a finitely generated abelian group, it is a direct product of finitely many cyclic groups. It will thus suffice to show that if N ⊲G and G/N is cyclic, then N is finitely generated. Since all finite index subgroups of G are finitely generated, we may assume that G/N is infinite cyclic. Let xN generate G/N. Given any generators {x1, …, xd} of G, we can write them in the form, where yi ∈ N, and then x, y1, …, yd generate G.