## Notes of David Kyed’s Cambridge lecture 18-05-2017

${\ell^2}$-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 ${\Gamma}$ act on vectorspace ${X}$. There are cohomology groups ${H^p(\Gamma,X)}$. 1-cocycles are maps ${c:\Gamma\rightarrow X}$ such that

$\displaystyle \begin{array}{rcl} c(\gamma\mu)=\gamma c(\mu)+c(\gamma). \end{array}$

1-coboundaries are ${\gamma\mapsto \gamma\xi-\xi}$, ${\xi\in X}$.

This coincides with Hochschild cohomology of the group algebra ${{\mathbb C}\Gamma}$ for the module ${X_\epsilon}$.

When ${X=\ell^2\Gamma}$, what is ${H^p(\Gamma,\ell^2\Gamma)}$ ? This is either 0 or infinite dimensional. One can attach a finite dimension to it: view it as a ${L\Gamma}$-module and take its ${dim_{L\Gamma}}$. What is this?

Assume that ${\Gamma}$ is finitely generated. Then cocycles are determined by their values on a finite generating set

$\displaystyle \begin{array}{rcl} (c(\gamma_1),\ldots,c(\gamma_n))\in \ell^2\Gamma^n. \end{array}$

It is a closed subspace, so one can project orthogonally to it. The projector is an ${n\times n}$ matrix with coefficients in ${L\Gamma}$, hence it has a finte trace. This defines

$\displaystyle \begin{array}{rcl} \beta_1^{(2)}(\Gamma)=\mathrm{dim}_{L\Gamma}\Gamma,\ell^2\Gamma)-1. \end{array}$

What are ${\ell^2}$-Betti numbers good for?

If ${\Gamma}$ is amenable, all ${\ell^2}$-Betti numbers vanish (Cheeger-Gromov). For free groups, it depends on rank.

The algebraic conjectures, like Kadison-Kaplansky’s, often follow from deeper ${\ell^2}$-Betti numbers conjectures.

2. Discrete quantum groups

A discrete quantum group is a Hopf, unital, star-algebra. It has a Haar trace ${h}$, which need not be a usual trace. We shall focus on quantum groups of Kac type, for which ${h}$ is a trace. This rules out quantum ${SU(2)}$.

Unfortunately, quantum groups tend not to act on spaces. The GNS construction allow to define an analogue of ${L^2({\mathbb C}\Gamma,h)}$, denoted by ${\ell^2\mathbb{G}}$, hence homology groups. When ${h}$ is a trace, ${\ell^2}$-Betti numbers are well defined.

2.1. Examples

Start with a compact Lie group (viewed as a dual of an inexistant discrete group). For ${O_n}$, let ${{\mathbb C}\hat O_n^+}$ be the ${\star}$ algebra generated by ${n}$ elements ${v_{ij}}$ subject to ${vv^\perp=1=v^\perp v}$.

For ${U_n}$, let ${{\mathbb C}\hat U_n^+}$ be the ${\star}$ algebra generated by ${n}$ elements ${u_{ij}}$ subject to ${u}$ and ${\bar u}$ being unitary.

For the symmetric group, viewed as permutation matrices, a similar algebra ${\hat S_n^+}$.

In all cases, the algebra can be view as a freed version of the group algebra. ${{\mathbb C}\hat U_n^+}$ is the non-commutative analogue of free groups. It behves much the same (it is a ${II_1}$ factor, it has rapid decay,…).

Open question. Is ${L\hat U_n^+\sim \hat U_m^+}$ for ${n\not m}$ ?

Banica proved that ${L\hat U_2^+\sim LF_2}$. ${\ell^2}$-Betti numbers have proved efficient in distinguishing these algebras.

3. Results

Vergnioux 2008 proved that ${\beta_1^{(2)}\hat O_n^+=0}$.

Collins-Hartel-Thom proved that ${\beta_p^{(2)}\hat O_n^+=0}$. This implies that ${\hat O_n^+}$ is not isomorphic to ${LF_m}$.

Theorem 1 (Raum 2016, Bichon-Kyed-Raum 2017) ${\beta_p^{(2)}\hat U_n^+=0}$ except for ${p=1}$ where it is equal to 1.

Vaes-Popa-Shlyakhtenko have defined ${\ell^2}$-Betti numbers for tensor categories. Such have representation categories ${Rep(\hat{\mathbb{G}})}$.

Theorem 2 (Kyed-Raum-Vaes-Valrekens 2017) ${\beta_p^{(2)}(\hat{ \mathbb{G}})=\beta_p^{(2)}(Rep (\hat{\mathbb{G}}))}$.

This might be the right definition, since the right hand side is always defined, whereas the left hand side makes sense only when ${h}$ is a trace.

4. Questions

What values can ${\ell^2}$-Betti numbers take ?

Tim Austin proved that uncountably many real numbers were ${\ell^2}$-Betti numbers. Since, Lukasz Grabowski (student of Schick) proved that every constructible real number is an ${\ell^2}$-Betti number.