-Betti numbers of universal quantum groups
Joint with Pichon, Arndt, Vaes,…
I spoke on the same subject in the same room in 2006. I think my understading has improved.
1. Infinite discrete groups
Let act on vectorspace . There are cohomology groups . 1-cocycles are maps such that
1-coboundaries are , .
This coincides with Hochschild cohomology of the group algebra for the module .
When , what is ? This is either 0 or infinite dimensional. One can attach a finite dimension to it: view it as a -module and take its . What is this?
Assume that is finitely generated. Then cocycles are determined by their values on a finite generating set
It is a closed subspace, so one can project orthogonally to it. The projector is an matrix with coefficients in , hence it has a finte trace. This defines
What are -Betti numbers good for?
If is amenable, all -Betti numbers vanish (Cheeger-Gromov). For free groups, it depends on rank.
The algebraic conjectures, like Kadison-Kaplansky’s, often follow from deeper -Betti numbers conjectures.
2. Discrete quantum groups
A discrete quantum group is a Hopf, unital, star-algebra. It has a Haar trace , which need not be a usual trace. We shall focus on quantum groups of Kac type, for which is a trace. This rules out quantum .
Unfortunately, quantum groups tend not to act on spaces. The GNS construction allow to define an analogue of , denoted by , hence homology groups. When is a trace, -Betti numbers are well defined.
Start with a compact Lie group (viewed as a dual of an inexistant discrete group). For , let be the algebra generated by elements subject to .
For , let be the algebra generated by elements subject to and being unitary.
For the symmetric group, viewed as permutation matrices, a similar algebra .
In all cases, the algebra can be view as a freed version of the group algebra. is the non-commutative analogue of free groups. It behves much the same (it is a factor, it has rapid decay,…).
Open question. Is for ?
Banica proved that . -Betti numbers have proved efficient in distinguishing these algebras.
Vergnioux 2008 proved that .
Collins-Hartel-Thom proved that . This implies that is not isomorphic to .
Theorem 1 (Raum 2016, Bichon-Kyed-Raum 2017) except for where it is equal to 1.
Vaes-Popa-Shlyakhtenko have defined -Betti numbers for tensor categories. Such have representation categories .
Theorem 2 (Kyed-Raum-Vaes-Valrekens 2017) .
This might be the right definition, since the right hand side is always defined, whereas the left hand side makes sense only when is a trace.
What values can -Betti numbers take ?
Tim Austin proved that uncountably many real numbers were -Betti numbers. Since, Lukasz Grabowski (student of Schick) proved that every constructible real number is an -Betti number.