A Kazhdan group with an infinite outer automorphism group

TL;DR

This paper constructs a specific Kazhdan group with Property (T) that also has an infinite outer automorphism group, expanding understanding of the automorphism structures of such groups.

Contribution

It demonstrates the existence of Kazhdan groups with infinite outer automorphism groups, specifically using semidirect products involving special linear groups.

Findings

The group $SL_n(K) times M_{n,m}(K)$ has Property (T) for $n \u2265 3$.

The outer automorphism group of this constructed group is infinite.

This provides new examples of Kazhdan groups with complex automorphism structures.

Abstract

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