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 $N$?
1 Answer
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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