Dehn function
You are handed a word that you know equals the identity, and asked to prove it — by inserting and cancelling copies of the defining relations until the word collapses. Some such proofs are short; some are agonizingly long. The Dehn function measures the worst case: given a word of length at most n that equals 1, how many applications of relations might you be forced to use? It is the cost, in relations, of certifying triviality.
Formally, for a finite presentation ⟨ S | R ⟩, a word w with w = 1 in the group can be written in the free group as a product of conjugates of relators; the area of w is the least number of relators (counted with multiplicity) in such an expression. The Dehn function is δ(n) = max{ area(w) : |w| ≤ n, w = 1 }. Up to a standard equivalence (the same as for growth functions), δ does not depend on the finite presentation and is a quasi-isometry invariant.
The Dehn function exactly governs the difficulty of the word problem: it is recursive if and only if the word problem is solvable, and its size measures the complexity. Linear Dehn function characterizes hyperbolic groups (Gromov); a quadratic bound holds for CAT(0) and automatic groups; nilpotent groups have polynomial Dehn functions; while groups like Baumslag-Solitar BS(1, 2) have exponential Dehn function, and there exist groups whose Dehn functions realize a rich ‘isoperimetric spectrum’ of exponents.
Z^2 = ⟨ a, b | [a, b] ⟩ has quadratic Dehn function: filling a loop of length n that traces out a rectangle in the grid takes on the order of n^2 commutator relators (the area of the rectangle). The Baumslag-Solitar group BS(1, 2) has exponential Dehn function.
Z^2 has quadratic Dehn function (n^2); BS(1, 2) needs exponentially many relators.
Geometrically the area of w is the minimal number of 2-cells (the relators) in a van Kampen diagram — a planar disk filling the loop w — so the Dehn function is an isoperimetric inequality: bounding area by a function of boundary length.