** Relative expanders **

**1. Coarse embeddings into Hilbert spaces **

In 1993, motivated by an approach suggested by Connes to Novikov conjecture, Gromov suggested to investigate coarse embeddings of groups in Hilbert space .

Definition 1Let be a metric space. A map is a coarse embedding if distances in the range and in the domain tend to infinity simultaneously (each is bounded by a function of the other).

**Question**: does every separable metric space admit such embeddings ?

A counterexample was soon obtained. It is based on old work of Per Enflo. Since anay metric space embeds isometrically in some , it is natural to look there for a counterexample.

Theorem 2 (Dranishnikov-Lafforgue-Yu)The sequence does not coarsely embed into .

Easy to discretize and organize into a graph. Drawback: degree of vertices is unbounded.

**2. Expanders **

In 2000, Gromov improved the example to get bounded degree graphs, using an expander.

Definition 3An expander is a sequence of finite graphs whose size tends to infinity, degree stays bounded, but Cheeger constant

stays bounded below.

**Example**. Large squares do not form an expander. Indeed, cutting in half requires only linear boundary. Trees do not either: cutting one edge suffices to split in half.

So it is not that easy to produce examples of expanders. Until 2000, the only known sources were

- Random graphs (Kolmogorov-Barzdin 1960’s, Pinsker 1973).
- Group theory (Margulis 1973).

**3. Expanders versus embeddings **

An alternate characterization of expanders was given by N. Alon in 1986.

Theorem 4 (Alon)The sequence is an expander iff it has a uniform spectral gap, i.e. for all and for every real valued function on ,

Corollary 5An expander does not coarsely embed into .

Indeed, a coarse embedding is -Lipschitz. The spectral gap provides us with a point such that

where is the degree. A positive proportion of points of satisfy . But, under a coarse embedding, the size of the inverse image of balls of fixed radius is bounded, contradiction.

Corollary 6A metric space that weakly contains an expander does not coarsely embed into .

*Weakly contains* means is -Lipschitz and inverse images of points represent a decaying proportion of points of .

**Question**. Conversely, if a bounded degree graph does not coarsely embed into , does it weakly contain an expander ?

**4. Expanders from group theory **

** 4.1. Property (T) **

Definition 7Let be a finitely generated group. Say has property (T) if every isometric affine action of on a Hilbert space has bounded orbits ( has a fixed point).

**Examples**. Infinite solvable (more generally, amenable) groups are not (T). Free groups are not (T). Lattices in higher rank simple Lie groups, e.g. , are (T).

Pick a generating system for . Take . Then get large and have degree bounded above by . I claim that have a uniform spectral gap.

** 4.2. A lemma **

**Fact**. Let be a property (T) group with generating system . There exists such that for every unitary representation and any cocycle ,

Proof is by contradiction. Otherwise, get sequence , with on but tends to infinity. Form the direct sum , set . Then the corresponding affine isometric action has unbounded orbits.

** 4.3. Spectral gap for **

View , with action on the right, as a unitary representation of . Take direct sum. Define cocycle by

for . Then

Furthermore, previous lemma yields

Averaging over yields the spectral gap (also known as Poincaré) inequality.

Note that the same argument would apply to amenable quotients of , instead of finite ones.