## Notes of Thibault Pillon’s lecture nr 2

1. Metric aspects

F\o lner’s criterium shows that amenability is a coarse equivalence invariant of groups. A-T-menability is not.

1.1. Yu’s Property A

From now on, we deal with metric spaces ${X}$. ${X}$ is assumed to be uniformly discrete. Yu pursued a purely (coarse) metric version of Bekka-Cherix-Valette’s theorem. He proposed a sufficient condition for coarse embeddability which, in the special case of groups, turned out to have a flavour similar to amenability and a-T-menability (it does not compare with either property).

Definition 1 (Yu 2000) A uniformly discrete metric space ${X}$ has property A if for all ${R}$, ${\epsilon>0}$, there exists a family ${\{A_x\}_{x\in X}}$ of subsets of ${X\times{\mathbb N}}$ such that

1. $\displaystyle \begin{array}{rcl} \frac{|A_x\Delta A_y|}{|A_x\cap A_y|}<\epsilon\quad\textrm{ whenever }d(x,y)

2. There exists ${S>0}$ such that ${A_x\subset B(x,S)\times{\mathbb N}}$.

The factor ${{\mathbb N}}$ is there to allow for sets with multiplicities.

In other words, (1) mimics F\o lner’s criterion. Sets are indexed by points in order to cope with the absence of equivariance.

Examples. Finite metric spaces have property A. Amenable groups have property A.

1.2. Trees have property A

Let ${T}$ be an infinite tree. Fix a root ${o}$. For ${n>0}$, set ${A_x^{(n)}=}$ vertices of the geodesic from ${x}$ to ${o}$ with multiplicity 1, until ${n}$ points have been counted. If ${\delta=d(x,o), put multiplicity ${n-\delta}$ on ${o}$. Then

$\displaystyle \begin{array}{rcl} |A_x\Delta A_y|\leq d(x,y),\quad |A_x\cap A_y|\geq n-2d(x,y). \end{array}$

1.3. Asymptotic dimension

Let ${X}$ be a metric space, ${\mathcal{U}}$ a cover of ${X}$. Its ${R}$-multiplicity is the maximum number of elements of ${\mathcal{U}}$ containing a given ball of radius ${R}$. ${\mathcal{U}}$ is uniformly bounded if there is a common bound for diameters of all elements of ${\mathcal{U}}$.

Definition 2 Say ${X}$ has asymptotic dimension ${\leq n}$ if for all ${R>0}$, there is a uniformly bounded cover of ${R}$-multiplicity ${\leq n+1}$.

Examples. ${{\mathbb Z}^n}$ has asymptotic dimension ${n}$.

${{\mathbb Z}^{(\infty)}}$, ${{\mathbb Z}\wr{\mathbb Z}}$ have infinite asymptotic dimension.

Gromov: Hyperbolic metric spaces have finite asymptotic dimension. However, it can be arbitrarily large.

Theorem 3 (Higson-Roe 2000) Finite asymptotic dimension implies Property A.

The proof is lengthy and not even included in my notes.

Theorem 4 (Yu 2000) Property A implies coarse embeddability in Hilbert space.

The construction is directly inspired from Bekka-Cherix-Valette. See below a more precise statement.

2. Quantitative Property A

Definition 5 (Tessera) Let ${X}$ be a uniformly discrete metric space. Let ${J:{\mathbb R}_+\rightarrow{\mathbb R}_+}$ e a nondecreasing function. Say that ${X}$ satisfies Property A with gauge ${J}$ and exponent ${p}$ if there exists a sequence of families ${(A_x^n)_{x\in X}}$ of subsets of ${X\times {\mathbb N}}$ such that

1. ${|A_x^n|\geq J(n)^p}$.
2. ${|A_x^n\Delta A_y^n|\leq C\,d(x,y)^p}$.
3. ${A_x^n}$ is contained in ${B(x,n)\times{\mathbb N}}$.

In other words, we want a quantitative control on the diameters of the sets in Yu’s Property A.

Theorem 6 From ${J}$, one cooks up a class of nondecreasing functions ${f}$ which are shown to be compression functions for Lipschitz coarse embeddings of ${X}$ into ${\ell^p}$.

Proof. Fix a base point ${o\in X}$. Embed ${X}$ in the ${\ell^p}$ direct sum of countably many copies of ${\ell^p(X\times{\mathbb N})}$ as follows. The ${n}$-component maps point ${x\in X}$ to

$\displaystyle \begin{array}{rcl} \frac{f(2^n)}{J(2^n)}(1_{A_x^{2^n}-1_{A_o^{2^n}}}. \end{array}$

This is indeed in ${\ell^p}$ provided

$\displaystyle \begin{array}{rcl} \int_{1}^{\infty}(\frac{f(t)}{J(t)})^{p}\frac{dt}{t}<\infty. \end{array}$

In fact, it is Lipschitz which compression ${f}$.

Examples. Spaces with subexponential growth have Property ${A(J,p)}$ for

$\displaystyle \begin{array}{rcl} J(t)=(\frac{t}{\log v(t))})^{1/p} \end{array}$

Doubling metric spaces have Property ${A(J,p)}$ with ${J(t)=t}$ and therefore coarsely embed into