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.


About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See
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