Is there an $N \in \mathbb{N}$ such that every finite index subgroup of $\mathrm{SL}\_3(\mathbb{Z})$ has a generating set of size $N$?



Is there a bound on the rank of finite index subgroup of SL_3(Z)?
No. Counterexamples are constructed by Sury and Venkataramana in (the answer is positive for *principal* congruence subgroups, but not otherwise):


MR1239806 (95a:20051) Reviewed 
Sury, B.(6-TIFR-SM); Venkataramana, T. N.(6-TIFR-SM)
Generators for all principal congruence subgroups of SL(n,Z) with n≥3. (English summary) 
Proc. Amer. Math. Soc. 122 (1994), no. 2, 355–358. 
20H05 


