Notes of Goulnara Arzhantseva’s first Oxford lecture 20-03-2017

Constructions of non ${C^*}$-exact groups

1. Motivation.
2. Coarse properties: graphs and groups.
3. Graphical presentations: from graphs to groups.
4. ${C^*}$-exact groups containing expanders.
5. ${C^*}$-exact groups having Haagerup property.

1. ${C^*}$-algebras

These are complete normed algebra with a conjugation ${\star}$. Basic examples are matrices, and more generally bounded operators ${B(\mathcal{H})}$ on a Hilbert space.

1.1. The reduced ${C^*}$-algebra of a group

The group algebra ${{\mathbb C} G}$ of a group is not a ${C^*}$-algebra since it is not complete. In fact, there are several useful completions. The simplest is obtained as follows. The left regular representation ${\lambda}$ of ${G}$ gives rise to an injective ${\star}$-homomorphism into ${B(\ell^2(G))}$. The reduced ${C^*}$-algebra of ${G}$ is the closure of the image of this homomorphism.

Example. ${C^*_{red}({\mathbb Z})=C(\mathbb{T})}$, the continuous functions on a circle.

1.2. Exactness

Definition 1 A ${C^*}$-algebra ${A}$ is exact if the functor ${A\otimes_{min}}$ is exact, i.e. maps short exact sequences to short exact sequences.

The minimal tensor product is the comletion of the algebraic tensor product forthe following norm: maximize norm of images under all unitary representations of the form ${\pi\otimes\rho}$.

One says that a ${C^*}$-algebra is nuclear if there is only one way to complete algebraic tensor products. It turns out that ${C^*_{red}(G)}$ is nuclear iff ${G}$ is amenable.

Kirchberg and Philips 2000 showed that separable exact ${C^*}$-algebras are exactly the subalgebras of nuclear ${C^*}$-algebras. Therefore Kirchberg raised the question of which are the ${C^*}$-exact groups, i.e. groups ${G}$ such that ${C^*_{red}(G)}$ is exact.

1.3. Examples

Hyperbolic groups. Amenable groups.

1.4. Coarse geometry motivation

Theorem 2 (Many authors, Higson-Roe, Anantharaman-Delaroche-Renault, Kirchberg-Wassermann,…) ${G}$ is exact iff ${G}$ is coarsely amenable.

This class of groups has nice properties: coarse Baum-Connes conjecture holds, hence Novikov conjecture, Kadison-Kaplansky conjecture, Borel conjecture hold.

Definition 3 A graph ${X}$ is coarsely amenable if ${\forall \epsilon>0}$, ${\exists S>0}$, ${\exists}$ map ${\xi:X\rightarrow \ell^1(X)}$ such that

1. ${\forall x}$, ${\|\xi_x\|=1}$.
2. ${\forall}$ edges ${xy}$, ${\|\xi_x-\xi_y\|<\epsilon}$
3. uniformly bounded supports.

Amenable graphs (hence groups) have it. Condition is stable under taking subgraphs (hence subgroups).

Exercise. Regular trees are coarsely amenable.

2. Non-coarsely amenable graphs

Lemma 4 (Willett 2011) Let ${X=\coprod X_n}$ be a coarse disjoint union of finite connected graphs. Assume that

1. Degrees of vertices are ${\geq 3}$ and bounded.
2. The girth of ${X_n}$ tends to infinity.

Then ${X}$ is not coarsely amenable.

Indeed, suppose we have maps ${\xi_x}$, i.e. functions ${\xi_x^n}$ with support in ${B(x,S)}$. Then

$\displaystyle \begin{array}{rcl} \sum_{\mathrm{edges}\,xy}\|\xi_x^n-\xi_y^n\|\leq D\epsilon |X_n|. \end{array}$

On the other hand,

$\displaystyle \begin{array}{rcl} |X_n|=\sum_{x,\,z}|\xi_x^n(z)|. \end{array}$

Hence there exists ${z_n\in X_n}$ such that

$\displaystyle \begin{array}{rcl} \sum_{\mathrm{edges}\,xy}|\xi_x^n(z_n)-\xi_y^n(z_n)|\leq D\epsilon\sum_{x\in X_n} |\xi_n(z_n)|. \end{array}$

Define

$\displaystyle \begin{array}{rcl} \psi^n=x\mapsto \xi_x^n(z_n). \end{array}$

Then ${\psi^n\in \ell^1}$, is has support in ${B(z_n,S)}$. Girth assumption allows to replace ${X_n}$ with a tree. Normalize ${\psi^n}$, get ${\phi^n}$ such that ${\|\phi^n\|_1=1}$ but ${\sum_{\mathrm{edges}\,xy}|\phi^n(x)-\phi^n(y)|\leq \epsilon}$. This sounds like amenability (in fact, it is stronger, go to level sets), and indeed cannot hold for trees, contradiction.

Beware. Trees are indeed coarsely amenable, so where is the cheat?

2.1. Scheme of construction

1. Find a suitable graph (as in Willett’s theorem). Where?
2. Coarsely embed it into a finitely generated group. How?
3. Conclude.

In 2003, Gromov gave the first construction of this kind. Ours will be simpler, really following the above scheme.

2.2. Box spaces

The suitable graphs will be box spaces, i.e. disjoint unions of Cayley graphs of finite quotients of a group ${G_0}$, put very far apart.

Take ${G_0}$ as a free group. This ensures that quotients have increasing girth.

Hence we get lots of non-coarsely amenable graphs, by Willett’s theorem.

2.3. Embedding graphs into groups

It will turn out that one can embed suitable graphs quasi-isometrically into groups (Finn-Sell 2015).

2.4. Non-expanders

Gromov’s examples are expanders. Ours are not.

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