** Presentation of the Fall School on Metric Embeddings **

**1. Basics **

Basic objects in this school are metric spaces. Here are a few definitions.

** 1.1. Classes of spaces **

Proper: closed balls are compact.

Locally finite: every ball has at most finitely many points in it.

Bounded geometry (also known as uniformly locally finite): an upper bound on the number of points in a ball depends only on radius.

** 1.2. Examples **

Graphs are equipped with the shortest path distance, where edges have length 1. In particular, Cayley graphs of finitely generated groups.

** 1.3. Maps **

Given a map between metric spaces, define the

**Expansion modulus** .

**Compression modulus** .

is *isometric* iff .

is *homothetic* (people sometimes call this isometric as well) if for some constant .

**Example**. A tripod cannot be isometrically embedded in Euclidean plane.

is *bi-Lipschitz* if , for some constants , , . When is a Banach space, one usually takes .

is a *uniform embedding* if and tends to 0 as tends to 0.

is a *coarse embedding* if and are finite and tends to infinity as tends to infinity.

is a *strong embedding* if it is uniform and coarse.

**Remark**. Coarse does not imply continuous nor injective.

**Example**. The integer part function is coarse.

is *quasi-isometric* or *coarse Lipschitz* if , for some constants , .

Example 1A subset in a metric space is an-skeletonif pairwise distances in are and every point of sits at distance from . When , one speaks of an-net. Then the nearest point map is quasisometric.

** 1.4. Corson-Klee Lemma **

Say a metric space is *metrically convex* if for all and all points , , there exists a point such that

**Example**. Graphs are metrically convex.

Lemma 1Let be metrically convex. Let have for some . Then

**2. Quantitative theory **

We are interested in the optimal behaviour of expansion and compression moduli among all maps between specific metric spaces.

** 2.1. Distorsion **

Denote by

The best distorsion of maps is called the \textrm{-distorsion} of .

** 2.2. Snowflaking exponent **

When , raising a metric to the power still gives a metric. This is the snowflaking operation.

Define

** 2.3. Compression exponent **

Introduced by Guentner and Kaminker. Define

No exponent on the right-hand side (i.e. for ), which makes it different from the snowflaking exponent. But .

**3. Why do we care ? **

Embedding a graph (network…) in a Banach space is a technique used in computer science

In geometric group theory, equivariant Hilbertian compression implies group is amenable.

In topology, coarse embeddability of the fundamental group into Hilbert space implies that Novikov’s conjecture holds for the space.