the growth of groups
Start at the identity of a group and ask: how many group elements can you reach in at most n steps using the generators? As n grows, this count grows too, and the speed at which it grows is a deep fingerprint of the group. The growth of a group measures how fast the number of elements within reach expands with the radius. A group whose count grows like a polynomial behaves very differently from one whose count explodes exponentially.
Fix a finitely generated group G with finite generating set S. The growth function is beta(n) = number of group elements g with word length |g|_S <= n; equivalently the number of vertices in the ball of radius n in the Cayley graph. The growth type classifies how beta behaves as n -> infinity, and the standard trichotomy is: polynomial growth, where beta(n) is bounded above and below by powers of n (like n^d for some d); exponential growth, where beta(n) grows like a^n for some a > 1; and the intermediate range in between. Z^d has polynomial growth of degree exactly d (a ball of radius n in the taxicab metric contains about n^d points). A free group on two generators has exponential growth — at each step you have roughly 3 new choices, so beta(n) is about 3^n. The growth type does not depend on the generating set: changing S only rescales beta within a quasi-isometry, so polynomial-versus-exponential is a genuine invariant.
Growth is a powerful and surprisingly rigid invariant. Gromov's polynomial-growth theorem is the landmark result: a finitely generated group has polynomial growth if and only if it is virtually nilpotent — it contains a nilpotent subgroup of finite index. This is astonishing because a purely geometric, counting condition (how fast balls grow) forces a precise algebraic structure (nilpotency up to finite index). On the other side, hyperbolic groups that are not virtually cyclic have exponential growth. The subtle honest point is the middle ground: Grigorchuk constructed groups of intermediate growth — faster than every polynomial but slower than every exponential — settling a question of Milnor and showing the trichotomy is real and the intermediate case genuinely occurs. So 'all groups are polynomial or exponential' is false.
Compare two groups. In Z^2 with standard generators, the ball of radius n is a diamond containing 2n^2 + 2n + 1 points — growth of polynomial degree 2, matching its dimension. In the free group F_2, the ball of radius n has 1 + 4(3^n - 1)/2 elements — at radius 0 there is 1, at radius 1 there are 5, and each further step multiplies the new boundary by about 3 — growth of exponential type. The first group is virtually nilpotent (indeed abelian); the second, having exponential growth, cannot be.
Z^2 grows polynomially (degree 2); the free group F_2 grows exponentially.
Not every group is polynomial or exponential. Grigorchuk's group has intermediate growth — strictly between every polynomial and every exponential — so the dichotomy fails; only polynomial growth has the clean algebraic characterization (virtually nilpotent, by Gromov).