Polynomial Growth of Nilpotent Groups

Theorem 4.1Nilpotent-by-finite groups have polynomial growth.

Proof By Proposition 2.5(c), we may assume that our group G is nilpotent. We employ induction on the Hirsch length h(G) of G. If h(G) = 1, then G is finite-by-(infinite cyclic)-by-finite, and so its growth type is the same as of ℤ, i.e. linear. Let G have a central series 1 = Gr+1 ≤ … ≤ G1 = G with cyclic factors, and let Gi = 〈Gi+1, xi〉, so that G = 〈x1, …, xr〉. If G/G2 is finite, then again it suffices to consider G2. We thus may assume that G/G2 is infinite, and then h(G2) = h(G) - 1, and the induction hypothesis applies to G2. Consider an element x ∈ G, written as a word of length n (or less) in the generators {xi}, say x = w1 = yi1 … yin, where each yi is either an xj or an. We are going to rewrite x in the form, for some integer e, where z ∈ G2. We start by looking for the first occurrence of x1 (or) that is to the right of another generator: say we have an occurrence of x2x1, and we replace that by the equal product x1x2[x2, x1]. If x2 was preceded by x3, we now have the product x3x1, which we replace by x1x3[x3, x1].