Uniform Kazhdan Constant for some families of linear groups

Abstract

Let $R$ be a ring generated by $l$ elements with stable range $r$. Assume that the group $E{L}_d(R)$ has Kazhdan constant ${\epsilon }_0>0$ for some $d>r$. We prove that there exist $\epsilon ({\epsilon }_0,l)>0$ and $k\in N$, s.t. for every $n\ge d$, $E{L}_n(R)$ has a generating set of order $k$ and a Kazhdan constant larger than $\epsilon$. As a consequence, we obtain for $S{L}_n(Z)$ where $n\ge 3$, a Kazhdan constant which is independent of $n$ w.r.t generating set of a fixed size.

To appear in J. of Alg

References in corpus (1)

• Finite Simple Groups as Expanders