On Kazhdan Constants Of Finite Index Subgroups In Sln(Z)

We prove that for any finite index subgroup G in SLn(Z), there exists k = k(n) is an element of N, epsilon = epsilon(Gamma) > 0, and an infinite family of finite index subgroups in Gamma with a Kazhdan constant greater than epsilon with respect to a generating set of order k. On the other hand, we prove that for any finite index subgroup G of SLn(Z), and for any epsilon > 0 and k is an element of N, there exists a finite index subgroup Gamma' <= Gamma such that the Kazhdan constant of any finite index subgroup in Gamma' is less than epsilon, 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 c/m, where c > 0 depends only on n. For a fixed n, this bound is asymptotically best possible.