The Growth of Nilpotent Groups
Avinoam Mann · Cambridge University Press eBooks · 2011
Polynomial Growth of Nilpotent Groups Theorem 4.1 Nilpotent-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 = G r +1 ≤ … ≤ G 1 = G with cyclic factors, and let G i = 〈 G i +1 , x i 〉, so that G = 〈 x 1 , …, x r 〉. If G / G 2 is finite, then again it suffices to consider G 2 . We thus may assume that G / G 2 is infinite, and then h ( G 2 ) = h ( G ) - 1, and the induction hypothesis applies to G 2 . Consider an element x ∈ G , written as a word of length n (or less) in the generators { x i }, say x = w 1 = yi 1 … yi n , where each y i is either an x j or an. We are going to rewrite x in the form, for some integer e , where z ∈ G 2 . We start by looking for the first occurrence of x 1 (or) that is to the right of another generator: say we have an occurrence of x 2 x 1 , and we replace that by the equal product x 1 x 2 [ x 2 , x 1 ]. If x 2 was preceded by x 3 , we now have the product x 3 x 1 , which we replace by x 1 x 3 [ x 3 , x 1 ].