Recursive area bounds and the word problem

A finite group presentation has a recursive upper bound on its Dehn function exactly when its word problem for a group is decidable. If the maximal relator length is $\ell$ and a word has length $n$, a diagram of area at most $N$ can be chosen with at most $\ell N+n$ edges. Enumerating all finite labelled planar diagrams within that bound gives a terminating test. Conversely, a word decider identifies the finitely many null words of length at most $n$; exhaustive finite diagram searches find their minimal areas, whose maximum computes the Dehn function.