Suppose I have a nice (e.g., word-hyperbolic? bi-automatic? automatic?) group and I want to know how big the smallest generating set is. Is that tractable (or, to put it more optimistically, what is the biggest class of groups for which it is tractable)? I am actually most interested in the question of whether there is a generating set of cardinality $2,$ but I suspect that is as hard as the general question.

**EDIT** What I *really* want to know is the answer for lattices (e.g., $SL(n, \mathbb{Z}),$) but that's probably not in any tractable class.