A bound for groups of linear growth
Let G be a finitely generated group and let f(n) be the number of elements which can be represented by words of length \(\leq n\) in the generators or their inverses. Suppose G is infinite, \(k>0\), and f(k)-f(k- 1)\(\leq k\). Set \(c=f(k)-f(k-1)\). It is shown that G has a subgroup isomorphic to \({\mathbb{Z}}\) of index \(\leq c\) and that this bound is sharp.
scientific article; zbMATH DE number 3935306