Generating pairs for finite index subgroups of sl(n, z)

TL;DR

This paper proves that for any finite index subgroup of SL(n, Z) with n ≥ 3, there exists a finite index subgroup generated by only two elements, advancing understanding of subgroup structures.

Contribution

It provides a positive answer to Lubotzky's question by showing all finite index subgroups of SL(n, Z) contain a 2-generated finite index subgroup.

Findings

Every finite index subgroup of SL(n, Z) contains a 2-generated finite index subgroup.

The result applies for all n ≥ 3.

It advances the understanding of subgroup generation in linear groups.

Abstract

Let $n \ge 3$. We positively answer a question of Lubotzky and prove that every finite index subgroup of SL(n, Z) contains a finite index subgroup which is generated by two elements.