On Kazhdan constants of finite index subgroups in $SL_n(\mathbb{Z})$

Abstract: We prove that for any finite index subgroup $\Ga$ in $SL_n(\mathbb{Z})$, there exists $k=k(n)\in\mathbb{N}$, $\ep=\ep(\Ga)>0$, and an infinite family of finite index subgroups in $\Ga$ with a Kazhdan constant greater than $\ep$ with respect to a generating set of order $k$. On the other hand, we prove that for any finite index subgroup $\Ga$ of $SL_n(\mathbb{Z})$, and for any $\ep>0$ and $k\in \mathbb{N}$, there exists a finite index subgroup $\Ga'\leq \Ga$ such that the Kazhdan constant of any finite index subgroup in $\Ga'$ is less than $\ep$, with respect to any generating set of order $k$. In addition, we prove that the Kazhdan constant of the principal congruence subgroup $\Gamma_n(m)$, with respect to a generating set consisting of elementary matrices (and their conjugates), is greater than $\frac{c}{m}$, where $c>0$ depends only on $n$. For a fixed $n$, this bound is asymptotically best possible.