On property (T) for automorphism groups of graph products

TL;DR

This paper characterizes when the automorphism group of a graph product of finite groups has Kazhdan's property (T), showing it occurs precisely when the underlying graph is complete.

Contribution

It provides a complete characterization of property (T) for automorphism groups of graph products of finite groups, linking it directly to the completeness of the defining graph.

Findings

Aut(G_Γ) has property (T) iff Γ is complete

Property (T) depends on the completeness of the graph

The result fully characterizes property (T) for these automorphism groups

Abstract

We show that the automorphism group of a graph product of finite groups $Aut(G_\Gamma)$ has Kazhdan's property (T) if and only if $\Gamma$ is a complete graph.