A cohomological description of property (T) for quantum groups

TL;DR

This paper establishes a cohomological framework for understanding property (T) in discrete quantum groups, linking it to first cohomology groups and demonstrating the vanishing of the first L^2-Betti number for such groups.

Contribution

It introduces a cohomological characterization of property (T) for quantum groups, extending classical theorems to the quantum setting.

Findings

Property (T) characterized via first cohomology groups

First L^2-Betti number of property (T) quantum groups vanishes

Provides a new cohomological perspective on quantum group rigidity

Abstract

We prove a Delorme-Guichardet type theorem for discrete quantum groups expressing property (T) of the quantum group in question in terms of its first cohomology groups. As an application, we show that the first L^2-Betti number of a discrete property (T) quantum group vanishes.