Notes of Thibault Pillon’s lecture nr 1

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

1. Banach-Tarski paradox

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.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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