Uniform Kazhdan Constant for some families of linear groups

Ouziel Hadad
Hebrew University

Let $R$ be a ring generated by $l$ elements with stable range $r$. Assume that the group $EL_d(R)$ has Kazhdan constant $\epsilon_0>0$ for some $d \geq r+1$. We prove that there exist $\epsilon(\epsilon_0,l) >0$ and $k \in \mathbb{N}$, s.t. for every $n \geq d$, $EL_n(R)$ has a generating set of order $k$ and a Kazhdan constant larger than $\epsilon$. As a consequence, we obtain for $SL_n(\mathbb{Z})$ where $n \geq 3$, a Kazhdan constant which is independent of $n$ w.r.t generating set of a fixed size.

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