Generating pairs for finite index subgroups of SL(n,Z)

Let n≥3. Lubotzky [2] asked if every finite index subgroup of SL(n,Z) contains a finite index subgroup which is generated by two elements. Venkataramana [11] proved that every finite index subgroup of SL(n,Z) contains a finite index subgroup which is generated by three elements. Since then it was widely believed that the answer to Lubotzky's question is positive and we show that this is indeed the case. In fact, we prove a stronger statement: for “almost every” element g∈SL(n,Z) there exists h∈SL(n,Z) such that [SL(n,Z):〈g,h^m〉]<∞ for every m≥1.