**Compression of embeddings**

Goal of talk is to explain why geometric group theorists are interested in metric embeddings.

**1. Coarse embeddings **

Definition 1Let be a metric space, a Banach space. Say a map is acoarse embeddingif

, are far apart iff , are far apart.

More precisely, if there exist two functions , , tending to , such that for all , ,

Remark 1If is a connected graph, one can take linear.

We are mainly interested in finitely generated groups. These are metric spaces, in a unique way up to coarse equivalence.

Example 1Baumslag-Solitar’s .

Represent it by matrices and . One can view these matrices as plane hyperbolic motions. The Cayley graph looks like the product of a degree tree and an interval.

Definition 2If is a group acting isometrically on , we say that is anequivariant coarse embeddingif there exists an (affine) isometric action of on such that for all and , .

Example 2The case of trees.

Let be a tree. Fix a base point . Define by

Then . So is a coarse embedding with .

Modify , to make it equivariant. Let denote the set of oriented edges. Define by

This is a cocycle: . Furthermore, is equivariant with respect to the natural representation of on . Fix vertex . Define an isometric action of on as follows. For , ,

Now define . Then is equivariant.

**2. Why do we care ? **

Guoliang’s Yu ‘s 2000 theorem cast topology into the coarse world.

Theorem 3Let be a finitely generated group. Assume that admits a coarse embedding into a Hilbert space. Then Novikov conjecture holds for .

Kasparov-Yu 2004: one can replace Hilbert space by uniformly convex Banach space.

Novikov conjecture is hard to formulate, but here a closely related conjecture, the *Borel stable conjecture*: Let , be closed manifolds with fundamental group . Assume that their universal covers are contractible. If and are homotopy equivalent, and are homeomorphic for large enough.

Question: can one remove ? Answer: yes, then it is called *Borel conjecture*, it is much more ambitious.

Higson and Kasparov later proved a much stronger result.

Theorem 4Let be a finitely generated group. Assume that admits a coarse embedding into a Hilbert space. Then Baum-Connes conjecture holds for .

Remark 2Some metric spaces do not admit any coarse embedding into Hilbert spaces.

E.g. spaces containing an expander family. This follows from Linial, London and Rabinovich’s paper mentioned this morning by Sanjeev Arora. Gromov was able to coarsely embed an expander family into the Cayley graph of a finitely generated group. This produces groups which do not admit any coarse embedding into Hilbert spaces.

Linial: can one use expanders arising from ? Answer; yes, since, for well chosen generators, these graphs are not only expanders, but have large girth, a fact which is required in Gromov’s construction.

**3. Compression **

The following terminology has been introduced by Kaminker and Guentner.

Definition 5Thecompression exponentof a map is the supremum of such that is linear and .

Thecompression of, , is the supremum of compressions of maps of to .

Similarly, define theequivariant compressionfor spaces with an isometric group action.

Example 3. (Kaminker and Guentner).

Proposition 6Yu’s property (A) embeds coarsely into Hilbert space.

Think of Yu’s property (A) as a weak form of amenability.

Proposition 7is amenable embeds coarsely into .

This is due to Naor and Peres, Bekka, Chérix and Valette.

**4. Permanence properties **

** 4.1. Wreath products **

S. Li, after Naor and Peres.

Theorem 8

and a similar statement for equivariant compression.

** 4.2. Free and amalgamated products **

Dreesen and Valette 2010.

Theorem 9Let where is finite, and has index at least in both and . Then

.

** 4.3. Extensions **

Kazhdan’s relative property (T) restricts lower bounds on equivariant compression of extensions in terms of equivariant compression of the factors. For instance, the natural semi-direct product has equivariant compression (in every Hilbert action, fixes a point) whereas both factors have positive equivariant compression.

In the other hand, Dreesen and Valette 2010 show

Theorem 10Let . Equip with the distance induced from .

If has polynomial growth, then .If is hyperbolic, then .

** References **