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 of the 2-sphere such that
- they constitute a partition of ;
- there are rotations and such that and (resp. and ) constitute partitions of .
This is a paradox since one has the impression to have doubled the .
Proof. Let , be irrational rotations with the smae angle, but different axes. One can prove that they generate a free subgroup of . Split the Cayley graph of that group into 4 subsets . (resp. ) consists of words beginning with an (resp. ). consists of words beginning with a , plus negative powers of . consists of words beginning with a , except negative powers of . Then
Use axiom of choice to pick a set of representatives of -orbits in . Let , and so on… This does the job, up to a countable set of axes (which I will ignore, for simplicity’s sake).
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 be a finitely generated group. Say is amenable if there exists a left-invariant mean on , i.e. a finitely additive (meaning not necessarily -additive) probability measure.
The existence of the partition 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. is amenable, but it is impossible to give an explicit mean. Ultrafilters can be used.
2. F\o lner’s criterion
Let be a group with finite generating set . Use corresponding word metric on . For a finite subset , let denote the -neigborhood of minus . Say is an -F\o lner set if
Equivalently (up to multiplicative constants),
Theorem 3 (F\o lner 1955) A finitely generated group is amenable if admits -F\o lner sets for all and .
The notion does not depend on the choice of generating system . In fact, it suffices to have a sequence of -F\o lner sets, , to prove amenability.
Example. On a free group on generators, every finite subtree of the Cayley graph has
This implies that every finite subset satisfies , so no F\o lner sets.
Example. In , is -F\o lner.
3. A-T-menability and proper actions
We consider isometric actions of a group by affine mappings of a Banach space . Such an action is proper if for some ,
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 on , oriented edges in the Cayley graph. For , let be the function on edges which is the indicator of the set of edges along the oriented geodesic from to , minus the indicator of the set of edges along the oriented geodesic from to . This is a cocycle, therefore it defines an affine action. Since 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 . Pick a sequence of -F\o lner subsets and set
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 far enough to have disjoint support. So action is proper.