On exponential growth and uniformly exponential growth for groups

摘要

Let G be a finitely generated group and let S be a finite generating set. The word length l_S(g) of an element g of G is the smallest integer n such that g is a product on n elements of S ∪ S~(-1). For each n ≥ 0 we define γs(n) = |{g ∈ G| l_S(g) ≤ n}|.