Kazhdan's Property T for Discrete Quantum Groups

TL;DR

This paper introduces a simple definition of property T for discrete quantum groups, establishes key properties, and provides the first example of a property T quantum group that is not derived from a classical group.

Contribution

It defines property T for discrete quantum groups, proves fundamental properties, and constructs the first non-group example using twisting.

Findings

Discrete quantum groups with property T are finitely generated and unimodular.

For I.C.C. quantum groups, property T is equivalent to Connes' property T.

First example of a property T quantum group not arising from a classical group.

Abstract

We give a simple definition of property T for discrete quantum groups. We prove the basic expected properties: discrete quantum groups with property T are finitely generated and unimodular. Moreover we show that, for "I.C.C." discrete quantum groups, property T is equivalent to Connes' property T for the dual von Neumann algebra. This allows us to give the first example of a property T discrete quantum group which is not a group using the twisting construction.