On Kazhdan constants of finite index subgroups in \(\text{SL}_n(\mathbb Z)\). (Q2888843)

The author proves that for any finite index subgroup \(\Gamma\) in \(\text{SL}_n(\mathbb Z)\), there exists \(k=k(n)\in\mathbb N\), \(\varepsilon=\varepsilon(\Gamma)>0\), and an infinite family of finite index subgroups in \(\Gamma\) with a Kazhdan constant greater than \(\varepsilon\) with respect to a generating set of order \(k\). On the other hand, he proves that for any finite index subgroup \(\Gamma\) of \(\text{SL}_n(\mathbb Z)\), and for any \(\varepsilon>0\) and \(k\in\mathbb N\), there exists a finite index subgroup \(\Gamma'\leq\Gamma\) such that the Kazhdan constant of any finite index subgroup in \(\Gamma'\) is less than \(\varepsilon\), with respect to any generating set of order \(k\). In addition, 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.