L²-Betti numbers for subfactors and rigid C*-tensor categories

Pioneered by Vaughan Jones in the 1980s, subfactors provide a powerful framework to encode quantum symmetries, with applications in various areas of mathematics, including knot theory. Any subfactor gives rise to a group-like structure, the so-called 'standard invariant'. Popa's work on the classifcation of subfactors with amenable standard invariant illustrates the need for a representation-theoretic framework to better understand the structure of this object. This framework, expressed in the language of rigid C*-tensor categories, was recently developed by Popa and Vaes. Using this new technology, they were able to generalise various properties of groups to the setting of general rigid C*-tensor categories, including L²-Betti numbers. These are important numerical group invariants, related to various structural properties of groups. During my PhD, I made several contributions to the theory of L²-Betti numbers in this context. Together with Kyed, Raum and Vaes, we related the L²-Betti numbers of representation categories of compact quantum groups with those of their discrete duals. As an application, we were able to compute the L²-Betti numbers of various discrete quantum groups. I later also proved a vanishing result for L²-Betti numbers of rigid C*-tensor categories, generalising a result of Bader, Furman and Sauer from the discrete group setting. In a joint article with Vaes, we also proved a spectral criterion for a rigid C*-tensor category to have property (T), generalising a theorem of Żuk. As an application, we gave the first 'genuinely quantum' examples of discrete quantum groups with property (T).

Publication year:2020

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