Dehn function

Summary

The Dehn function of a finitely presented group measures how many relators are needed to prove that a word of length $n$ equals the identity. Up to a standard equivalence it does not depend on the presentation, and it is a quasi-isometry invariant. It is a geometric measure of the difficulty of the word problem. The word problem is solvable exactly when the Dehn function is bounded by a computable function. A group is hyperbolic if and only if its Dehn function is linear.