** Coarse embeddings and how to avoid them **

We want to study coarse embeddings for finitely generated groups and . A subgroup is coarsely embedded, hence this is a coarse geometric analogue of studying subgroups.

**Example**. Although cannot be a subgroup of a hyperbolic group, it can be coarsely embedded in hyperbolic 3-manifold groups.

**Question 1**. Does there exist a coarse embedding of into a hyperbolic group?

**Example**. Does there exist an infinite sequence such that has a coarse embedding into iff ? Yes, use asymptotic dimension.

**Question 2**. Does there exist an infinite sequence such that has a coarse embedding into iff ? Not so obvious, due to lack of invariants.

**1. Local decompositions **

Let be a finite graph. Cutting it into 2 pieces leads to Cheeger’s constant. The -Poincare constant is an analytic version of Cheeger’s constant.

For , equals Cheeger’s constant up to a factor of 2.

**2. Asymptotic decompositions **

Definition 1The -Poincare profile of an infinite graph is

The joyful stuff is that -Poincare profile is monotone under coarse embeddings of infinite graphs with bounded degrees.

** 2.1. What is being measured? **

Theorem 2 (Hume 2015)If , coincides with Benjamini-Schramm-Timar’s separation profile.

Proposition 3 (Hume-MacKay-Tessera)If , depends on volume growth only,where

Since increases with , we see that interpolates between connectivity and growth.

** 2.2. Examples **

1. For a 4-regular tree, is bounded, , .

It follows that detects trees.

Proposition 4 (Hume-MacKay)is bounded iff is qi to a tree.

2. contains an expander iff does not tend to 0, for any finite . So detects expanders.

3. Let . Then .

Proposition 5 (Hume-MacKay-Tessera)If has finite linearly conterolled asymptotic dimension, then

for all .

This does not follow from monotonicity, due to the presence of multiplicative constants in monotonicity.

**Question**. What about ? This has (nonlinearly controlled) asymptotic dimension 2 but not finite linearly controlled asymptotic dimension (Nowak).

4. detects polynomial growth. Indeed, has polynomial growth iff there exists such that for all finite , .

Theorem 6 (Hume-MacKay-Tessera)If is virtually nilpotent with polynomial growth of degree , then

for all .

Corollary 7Rank 1 symmetric spaces satisfy where is the conformal dimension.

Indeed, horospheres gives an upper bound. Lower bounds are harder, we get one only for real hyperbolic space.

Theorem 8For -dimensional real hyperbolic space,

Using the Bonk-Schramm embedding, it follows that every hyperbolic graph has polynomial for some , for large.

**Question**. What is the optimal ?

Theorem 9For Bourdon buildings of conformal dimension ,

So we clearly see how switches from a behaviour governed by connectivity to a behaviour governed by growth.

**3. Back to coarse embeddings **

Since conformal dimensions of Bourdon buildings accumulate near 1, we get an answer to Question 2 above.

For Question 1, we use lamplighter group , for which

The critical subgraphs are where lit lamps and the lamplighter are in . The optimal partition of is

If follows that lamplighter group cannot be coarsely embedded into any hyperbolic group.

However, there is a coarse embedding into . Indeed, both are horospheres in products of trees, for which a Busemann function preserving coarse embedding exists. This answers Question 1.