If $G = \langle X | R \rangle$ is a $\delta$-hyperbolic group presentation, then Dehn's algorithm provides a linear time solution to the word problem, but the linear constant is horribly exponential in $\delta$ since the Dehn presentation consists of all words of length $8 \delta$ equal to the identity.

Are there any known lower bounds on the number of relators required to make Dehn's algorithm solve the word problem? In other words, does anyone know a family of groups for which the required number of Dehn relators is exponential in either the size of the original group's presentation or the original groups $\delta$.