## Notes of Ana Khukhro’s lecture

Wall structures and coarse embeddings

Lots of spaces have Property A, great! This provides many coarse embeddings in Hilbert space. Nevertheless, there exist spaces which coarsely embed but do not satisfy Property A. I will describe the best known obstruction to Property A, expansion. This also prevents Hilbertian embeddings. Then I will describe an other mechanism for the construction if embeddings, spaces with walls.

1. Is Property A equivalent to coarse embeddability in Hilbert space ?

1.1. Expansion

A sequence of finite graphs is called an expander if

• Size tends to infinity.
• Valency stays uniformly bounded.
• Cheeger constant stays bounded below.

Note the forthcoming conference in Neuchatel (Switzerland), december 1-5: Expanders everywhere!

Given such a sequence, form a metric space, their disjoint union, by requiring that distance between distinct graphs be larger than their diameters.

Fact. An expander does not have Property A and does not coarsely embed in Hilbert space.

1.2. A space which does not have Property A but does coarsely embed in Hilbert space

The following example is due to P. Nowak (2002). One deals with embeddings in ${L^1}$ (from the point of view of coarse embeddings, this is equivalent to ${L^2}$).

Let ${F}$ be a finite group, with generating set ${F}$ itself, embed it bi-Lispchitaly in ${L^1}$. Consider direct sum ${F^n}$ with obvious generating system. This embeds in ${L^1}$ with the same distorsion (direct sum of embeddings), so the disjoint union embeds as well.

When ${F={\mathbb Z}_2}$, we are dealing with Hamming cubes, we shall see below an alternating embedding in ${\ell^1}$.

Nowak verifies that Property A fails. This requires some effort (not included in my notes).

2. Wall spaces

2.1. Definition

Let ${\Gamma}$ be a graph. A wall on ${\Gamma}$ is a subset of edges whose removal yields exactly two connected components. A wall structure on ${\Gamma}$ is a set of walls ${\mathcal{W}}$ such that each edge is contained in exactly one wall. Notation :

$\displaystyle \begin{array}{rcl} W(x|y)=\{\textrm{walls separating }x\textrm{ and }y\}. \end{array}$

The wall (pseudo) metric is defined by

$\displaystyle \begin{array}{rcl} d_w(x,y)=|W(x|y)|. \end{array}$

When ${d_w}$ is a metric, it embeds isometrically in ${\ell^1(\mathcal{W})}$ as follows. Fix an origin ${o}$. Map vertex ${x}$ to indicator of ${W(x|o)}$.

Example. The graph metric on a tree is a wall metric. Each single edge is a wall. Take all of them.

Example. Hamming cube. The set of edges in one direction is a wall. Take all of them. The resulting metric equals the Hamming metric.

2.2. Bounded geometry

Ultimately, we would like to provide examples of finitely generated groups without property A (but which embed coarsely in ${\ell^1}$. For this, we would like to proide a graph counterexample and then embed it is a group. This requires the graph to have bounded geometry.

Nowak’s example does not have this property (valency tends to infinity), so we turn to different constructions.

3. Residual finiteness

Say a group ${G}$ is residually finite if every non trivial element has a non trivial image under some homomorphism to some finite group. Equivalently, the intersection of all finite index normal subgroups of ${G}$ is ${\{e\}}$.

This often happens.

Examples. Free groups, abelian groups, linear groups are residually finite. It is an open question wether all hyperbolic groups are residually finite.

Definition 1 Given a sequence of nested finite index normal subgroups

$\displaystyle \begin{array}{rcl} G>N_1>N_2\cdots,\quad \bigcap N_i=\{e\}, \end{array}$

the box space is the coarse disjoint union of ${Cay(G/N_i,S_i)}$, where ${S_i}$ is the projection to ${G/N_i}$ of a fixed generating system of ${G}$.

Theorem 2 (Guentner) The box space of a group ${G}$ has property A if and only if ${G}$ is amenable.

Indeed, F\o lner sets can be used for the box space. Conversely, the F\o lner sets of the box space can be averaged to provide F\o lner sets for ${G}$.

3.1. Derived series

Given a finitely generated group ${G}$, the derived ${m}$-series ${N_i}$ is defined recursively by ${N_1=G}$ and

$\displaystyle \begin{array}{rcl} N_{i+1}=[N_i,N_i]N_i^m. \end{array}$

The idea is to kill commutators and ${m}$-powers, i.e. to provide the largest quotient ${N_i/N_{i+1}}$ which is abelian with order ${m}$. Then ${N_i/N_{i+1}}$ is a direct sum of cyclic groups ${{\mathbb Z}_m}$.

Theorem 3 (Arzhantseva-Guentner-Spakula 2012) Let ${F}$ be a non abelian free group. Let ${\{N_i\}}$ be its derived 2-series. Let ${X}$ be the corresponding box space. Then ${X}$ embeds coarsely in ${\ell^2}$.

3.2. Proof

View ${F/N_{i+1}}$ as a cover of ${F/N_i}$. This will provide us with a wall structure on ${F/N_{i+1}}$ whose wall metric is close to the origin metric.

The covering map is constructed explicitely by lifting a maximal subtree of ${Cay(G/N_i,S_i)}$ en specifying how the copies are glued together. The fibers are cubes, which are spaces with walls.

The walls of ${Cay(G/N_{i+1},S_{i+1})}$ are inverse images, under the covering map, of edges of ${Cay(G/N_i,S_i)}$. The wall metric is smaller than the graph metric, and they coincide below the girth. Since girth tends to infinity, ther wall metric gets closer and closer to the graph metric. This yields the ${\ell^1}$ embedding.

3.3. Generalization

What we really use is the existence of embeddings for the disjoint union of quotient groups, not that much the wall structure.