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).