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.
UPDATE It is, in fact, known that $SL(n, \mathbb{Z})$ itself is generated by $2$ elements (Hua+Reiner wrote down a generating set with three elements in 1949, as did M. Conder et al in 1992, for $SL(3, \mathbb{Z})$, but Stanton M. Trott did it with two generators in 1962). The generators are:
\begin{pmatrix} 1 & 0 & 0 & \dots & 0 \\ 1 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ . & . & . & \dots & .\\ 0 & 0 & 0 & \dots & 1 \end{pmatrix}$\begin{pmatrix}
1 & 0 & 0 & \dots & 0 \\
1 & 1 & 0 & \dots & 0 \\
0 & 0 & 1 & \dots & 0 \\
. & . & . & \dots & .\\
0 & 0 & 0 & \dots & 1
\end{pmatrix}$
and
\begin{pmatrix} 0 & 1 & 0 & \dots & 0\\ 0 & 0 & 1 & \dots & 0\\ . & . & . & \dots & .\\ 0 & 0 & 0 & \dots & 1\\ (-1)^n & 0 & 0 & \dots & 0 \end{pmatrix}$\begin{pmatrix}
0 & 1 & 0 & \dots & 0\\
0 & 0 & 1 & \dots & 0\\
. & . & . & \dots & .\\
0 & 0 & 0 & \dots & 1\\
(-1)^n & 0 & 0 & \dots & 0
\end{pmatrix}$
