TITLE: size of smallest generating set of a group
QUESTION [21 upvotes]: 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: 
$\begin{pmatrix} 
1 & 0 & 0 & \dots & 0 \\ 
1 & 1 & 0 & \dots & 0 \\ 
0 & 0 & 1 & \dots & 0 \\ 
. & . & . & \dots & .\\ 
0 & 0 & 0 & \dots & 1 
\end{pmatrix}$
and 
$\begin{pmatrix}
0 & 1 & 0 & \dots & 0\\
0 & 0 & 1 & \dots & 0\\
. & . & . & \dots & .\\
0 & 0 & 0 & \dots & 1\\
(-1)^n & 0 & 0 & \dots & 0
\end{pmatrix}$

REPLY [5 votes]: There is no upper bound for finite index subgroups of $SL_n({\mathbb Z})$. That is, given any integer $k$, there exists a finite index subgroup of $SL_n({\mathbb Z})$ which needs at least $k$ generators. I do not know to whom this is originally due, but it is a remark of Rapinchuk that if you pull the centre of, say $SL_{2n}({\mathbb Z}/m {\mathbb Z})$ where $m$ is the product of $k$ distinct primes back to $SL_{2n}({\mathbb Z}) $, you get a finite index subgroup of $SL_{2n}({\mathbb Z})$ whose abelianisation has  $({\mathbb Z}/2{\mathbb Z})^k$ as a quotient and hence is at least $k$ generated. 
On the other hand, a finite index subgroup of $SL_n({\mathbb Z})$, contains a smaller finite index subgroup which is generated by 3 elements; this result is true for any higher rank non-uniform arithmetic group in a semi-simple linear group (and is due to Ritumoni Sarma and Venkataramana)