A Kazhdan group with an infinite outer automorphism group

Abstract

D. Kazhdan has introduced in 1967 the Property (T) for local compact groups. In this article we prove that for $n\ge 3$ and $m\in \mathbb{N}$ the group $S{L}_n(\text{K})⋉{\mathcal{M}}_{n,m}(\text{K})$ is a Kazhdan group having the outer automorphism group infinite.