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

We prove that for any finite index subgroup $\text{Ga}$ in $S{L}_n(\mathbb{Z})$, there exists $k=k(n)\in \mathbb{N}$, $\text{ep}=\text{ep}(\text{Ga})>0$, and an infinite family of finite index subgroups in $\text{Ga}$ with a Kazhdan constant greater than $\text{ep}$ with respect to a generating set of order $k$. On the other hand, we prove that for any finite index subgroup $\text{Ga}$ of $S{L}_n(\mathbb{Z})$, and for any $\text{ep}>0$ and $k\in \mathbb{N}$, there exists a finite index subgroup ${\text{Ga}}^{\mathrm{\prime }}\le \text{Ga}$ such that the Kazhdan constant of any finite index subgroup in ${\text{Ga}}^{\mathrm{\prime }}$ is less than $\text{ep}$, with respect to any generating set of order $k$. In addition, we prove that the Kazhdan constant of the principal congruence subgroup ${\mathrm{\Gamma }}_n(m)$, 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.