growth function
Stand at the identity of a group and ask: how many elements can I reach in at most n steps? Take one step — count the neighbors. Take two steps — count everything within reach. The tally as a function of the radius n is the growth function. It records how fast the group ‘opens up’ around you: a thin group barely grows, a bushy free group explodes.
Fix a finitely generated group G with finite symmetric generating set S. The growth function is β_S(n) = |{ g in G : |g|_S ≤ n }|, the number of elements in the closed ball of radius n in the word metric. (Some authors use the sphere-counting variant, counting elements of word length exactly n.) The function depends on S, but its asymptotic behavior does not, which is what makes it a useful invariant.
Two growth functions f and g are called equivalent if each is dominated by a linear rescaling of the other: f(n) ≤ A g(Bn + C) + D and symmetrically. Under this relation the equivalence class of β_S is independent of the generating set and is a quasi-isometry invariant. This class — the group's growth type — is the content captured separately under growth rate.
For Z^2 with the standard generators, the ball of radius n is a diamond with about 2n^2 + 2n + 1 lattice points, so β(n) grows like n^2. For the free group F_2 on two generators, β(n) = 1 + 4·(3^n - 1)/2, which grows like 3^n.
Z^2 grows polynomially (n^2); the free group F_2 grows exponentially (3^n).