** -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.

** 2.1. Examples **

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.

**3. Results **

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.

**4. Questions **

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.