## Notes of Goulnara Arzhantseva’s third Oxford lecture 22-03-2017

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

Although we seem to construct counterexamples, observe that the technique I explained is used for positive results.

1. Non ${C^*}$-exact groups with Haagerup property

Amenability implies Coarse amenability which implies Coarse embeddability.

In turn, Amenability implies Haagerup property which implies Coarse embeddability.

But there is no logical relation between Haagerup property and Coarse amenability. We show this by an example.

1.1. Wall structure

With Guentner and Spakula, we construct wall structures on graphs.

Let ${p:\tilde X_n\rightarrow X_n}$ be the finite characteristic mod 2 covering of graph ${X_n}$ (i.e. induced by ${H^1(X_n,{\mathbb Z}/2{\mathbb Z})}$). If ${X_n}$ is 2-connected, ${\tilde X_n}$ has a wall structure, where walls are inverse images of edges of ${X_n}$. The corresponding wall distance is equal to the graph metric at scales less that the girth of ${X_n}$. Since girth tends to infinity, this implies that ${\coprod\tilde X_n}$ with wall distance is coarsely equivalent to ${\coprod X_n}$.

We apply this construction to ${X_n}$ obtained by iteratively taking covers induced from the normal subgroup generated by squares. At each step, the graph is a Cayley graph, and small cancellation labellings pass the step provided small cancellation is meant up to self graph automorphisms.

Walls are on the graph. With Osajda, we next get walls on the group, using the lacunary walling condition. This is inspired from Wise’s wall structure for classical small cancellation groups. It works because of dg-boundedness. It turns out that the obtained wall distance is bi-Lipschitz equivalent to word metric on group.

It is standard that groups acting properly on wall spaces have Haagerup property.

2. Non ${C^*}$-exact groups containing no expanders

With Tessera. Let ${W}$ be the semi-direct product

$\displaystyle \begin{array}{rcl} W={\mathbb Z}/2{\mathbb Z}\wr_G H:=\bigoplus_{G}{\mathbb Z}/2{\mathbb Z} \times H \end{array}$

where ${G}$ is the special Gromov monster, and ${H}$ the Haagerup monster which come with a homomorphism onto ${G}$. Due to the big abelian part, ${W}$ does not contain expanders.

3. Open questions

Can one embed Lafforgue’s expander into a group? This would provide groups non-embeddable in certain Banach spaces. Lack of dg-boundedness.

Is there a finitely presented true coarse monster group?

Thompson group is Haagerup. Is it coarsely amenable ?

Grigorchuk group’s is amenable, thus it embeds in Hilbert space. With which compression?