The Dehn function of a finitely presented group is a quasi-isometry invariant that geometrically characterizes the solvability of the word problem. If the word problem is solvable, the Dehn function leads to an upper bound on the complexity of the solution.
However, some groups have Dehn functions which grow super-exponentially but admit efficient polynomial time solutions to their word problems.

I will examine some of the techniques used to find more efficient solutions to the word problem of a group with a large Dehn function. I will discuss the Hydra groups of Dison & Riley whose Dehn functions grow like Ackermann functions and will explain polynomial time solutions to their word problems (which is joint work with Dison & Riley).