growth rate
Two groups might both grow without bound, yet one fills out gently like the area of an expanding disk while the other doubles at every step like a branching tree. The growth rate names this qualitative tempo. It throws away the exact counts and keeps only the regime: does the ball of radius n grow like a polynomial in n, like an exponential, or somewhere strictly in between?
A finitely generated group has polynomial growth if β(n) ≤ C n^d for some constants C, d; exponential growth if β(n) ≥ a^n for some a > 1 (equivalently the limit of β(n)^(1/n) exceeds 1); and intermediate growth if it is faster than every polynomial yet slower than every exponential. These three classes exhaust the possibilities and are each invariant under quasi-isometry. The exponential growth rate lim β(n)^(1/n) itself does depend on the generating set, but whether it exceeds 1 does not.
Two profound theorems frame this invariant. Gromov's theorem (1981) states that a finitely generated group has polynomial growth if and only if it is virtually nilpotent — an astonishing bridge from coarse geometry to algebraic structure. And Grigorchuk's group (1984) settled Milnor's question by exhibiting a group of genuine intermediate growth, so the trichotomy is not a dichotomy: the middle ground is inhabited.
Z^d has polynomial growth of degree d; the discrete Heisenberg group (integer matrices [1, a, c; 0, 1, b; 0, 0, 1]) is nilpotent and has polynomial growth of degree 4, not 3. Any group containing a free subgroup of rank 2, such as SL(2, Z), has exponential growth.
Heisenberg grows like n^4 — polynomial growth need not match the obvious dimension.
Nilpotent groups have polynomial growth of degree given by the Bass-Guivarc'h formula, a weighted sum over the ranks of the lower central series quotients.