Kazhdan constants for two-element generating sets of $\mathrm{SL}_n(\mathbb Z)$

Abstract

For $n\geq3$, the group $\mathrm{SL}_n(\mathbb Z)$ has property $(T)$, so every finite generating set has a positive Kazhdan constant. However, it was recently shown that the infimum of these constants over all finite generating sets is $0$. This naturally raises the question of whether the infimum remains zero when the cardinality of the generating sets is bounded in advance. We prove that the infimum of the Kazhdan constants of $\mathrm{SL}_n(\mathbb Z)$ over all two-element generating sets is $0$ for every $n\geq3$.