Property (T) discrete quantum groups and subfactors with triangle presentations

Abstract

Kazhdan's property (T) has been studied for several discrete group-like structures, including standard invariants of Jones' subfactors and discrete quantum groups. We prove a Zuk-type spectral gap criterion for property (T) in this setting. This allows us to construct a family of property (T) discrete quantum groups, as well as subfactor standard invariants, given by generators and a triangle presentation. These are the first examples of property (T) discrete quantum groups that are not directly related to a discrete group with property (T).

arXiv commentv2: Minor changes, final version, to appear in Advances in Mathematics

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Cited by in corpus (4)

• Quantum automorphism groups of connected locally finite graphs and quantizations of discrete groups
• Traces On Diagram Algebras I: Free Partition Quantum Groups, Random Lattice Paths And Random Walks On Trees
• Triangle presentations and tilting modules for ${\text{SL}}_{2k+1}$
• $L^2$-Betti numbers of $C^{\ast }$-tensor categories associated with totally disconnected groups