Notes of Thibault Pillon’s lecture nr 1

F\o lner type sets, Property A and coarse embeddings

Theorem 1 (Hausdorff 1914, Banach-Tarski 1924) There exists 4 disjoint subsets ${A_1,A_2,B_1,B_2}$ of the 2-sphere ${S^2}$ such that

1. they constitute a partition of ${S^2}$ ;
2. there are rotations ${\alpha}$ and ${\beta}$ such that ${A_1}$ and ${\alpha(A_2)}$ (resp. ${B_1}$ and ${\beta(B_2)}$) constitute partitions of ${S^2}$.

This is a paradox since one has the impression to have doubled the ${S^2}$.

Proof. Let ${\alpha}$, ${\beta}$ be irrational rotations with the smae angle, but different axes. One can prove that they generate a free subgroup ${G}$ of ${SO(3)}$. Split the Cayley graph of that group into 4 subsets ${\mathcal{A}_1,\mathcal{A}_2,\mathcal{B}_1,\mathcal{B}_2}$. ${\mathcal{A}_1}$ (resp. ${\mathcal{A}_2}$) consists of words beginning with an ${\alpha}$ (resp. ${\alpha^{-1}}$). ${\mathcal{B}_1}$ consists of words beginning with a ${\beta}$, plus negative powers of ${\beta}$. ${\mathcal{B}_2}$ consists of words beginning with a ${\beta}$, except negative powers of ${\beta}$. Then

$\displaystyle \mathcal{A}_1 \cup\alpha\mathcal{A}_2=\mathcal{B}_1 \cup\beta\mathcal{B}_2.$

Use axiom of choice to pick a set ${R}$ of representatives of ${G}$-orbits in ${S^2}$. Let ${A_1=\mathcal{A}_1 R}$, and so on… This does the job, up to a countable set of axes (which I will ignore, for simplicity’s sake).

1.1. Amenability

Von Neumann understood that Banach-Tarski’s paradox stemmed from a property of free groups, which he abstracted as follows.

Definition 2 (von Neumann 1929) Let ${G}$ be a finitely generated group. Say ${G}$ is amenable if there exists a left-invariant mean on ${G}$, i.e. a finitely additive (meaning not necessarily ${\sigma}$-additive) probability measure.

The existence of the partition ${\mathcal{A}_1,\mathcal{A}_2,\mathcal{B}_1,\mathcal{B}_2}$ of the free group make it impossible for it to admit a left-invariant mean. In fact, Tarski showed that a group admit a paradoxical decomposition iff it is non amenable.

Examples. Finite groups are amenable, free groups are not. ${{\mathbb Z}}$ is amenable, but it is impossible to give an explicit mean. Ultrafilters can be used.

2. F\o lner’s criterion

Let ${G}$ be a group with finite generating set ${S}$. Use corresponding word metric on ${G}$. For a finite subset ${A\subset G}$, let ${\partial_r A}$ denote the ${r}$-neigborhood of ${A}$ minus ${A}$. Say ${A}$ is an ${(r,\epsilon)}$-F\o lner set if

$\displaystyle \begin{array}{rcl} \frac{|\partial_r A|}{|A|}<\epsilon. \end{array}$

Equivalently (up to multiplicative constants),

$\displaystyle \begin{array}{rcl} \frac{|gA\Delta A|}{|A|}<\epsilon\quad\forall g\in B(r). \end{array}$

Theorem 3 (F\o lner 1955) A finitely generated group is amenable if admits ${(r,\epsilon)}$-F\o lner sets for all ${r>0}$ and ${\epsilon>0}$.

The notion does not depend on the choice of generating system ${S}$. In fact, it suffices to have a sequence of ${(1,\epsilon_n)}$-F\o lner sets, ${\epsilon_n\rightarrow 0}$, to prove amenability.

Example. On a free group on ${k}$ generators, every finite subtree ${A}$ of the Cayley graph has

$\displaystyle \begin{array}{rcl} |\partial_1 A|=(2k-2)|A|+2. \end{array}$

This implies that every finite subset satisfies ${|A|\leq \frac{1}{2k-2}|\partial A|}$, so no F\o lner sets.

Example. In ${{\mathbb Z}}$, ${[-n,n]}$ is ${(r,\frac{r}{n})}$-F\o lner.

3. A-T-menability and proper actions

We consider isometric actions ${\alpha}$ of a group ${G}$ by affine mappings of a Banach space ${B}$. Such an action is proper if for some ${\xi\in B}$,

$\displaystyle \begin{array}{rcl} |\alpha(g)\xi|\rightarrow+\infty\quad \textrm{as}\quad |g|\rightarrow+\infty \textrm{ in }G. \end{array}$

Definition 4 (Gromov 1988) A group is a-T-menable if it admits a proper affine isometric action on Hilbert space.

This is a strong negation of Kazhdan’s property (T), whence the terminology. It turns out to be equivalent to a contemporary notion, Haagerup’s property in operator algebras.

3.1. Example: free group

Start with the unitary representation of ${G}$ on ${\ell^2(E)}$, ${E=}$ oriented edges in the Cayley graph. For ${g\in G}$, let ${b(g)}$ be the function on edges which is the indicator of the set of edges along the oriented geodesic from ${e}$ to ${g}$, minus the indicator of the set of edges along the oriented geodesic from ${g}$ to ${e}$. This is a cocycle, therefore it defines an affine action. Since ${|b(g)|}$ tends to infinity, the action is proper.

So free groups are a-T-menable.

3.2. The case of amenable groups

Theorem 5 (Bekka-Cherix-Valette, 1993) Amenable groups are a-T-menable.

Proof. Take the orthogonal direct sum of counably many copies of ${\ell^2(G)}$. Pick a sequence of ${(1,\frac{1}{n^2})}$-F\o lner subsets ${F_n}$ and set

$\displaystyle \begin{array}{rcl} \xi_n=\frac{1}{|F_n|}1_{F_n}. \end{array}$

Define

$\displaystyle \begin{array}{rcl} b(g)=\bigoplus_{n=1}^{\infty}g\xi_n -\xi_n. \end{array}$

This is a cocycle (by construction), so it defines an affine action. Its squared norm is finite and tends to infinity because far away elements push the first ${\xi_n}$ far enough to have disjoint support. So action is proper.