## Notes of David Hume’s Southampton lecture 30-03-2017

Coarse embeddings and how to avoid them

We want to study coarse embeddings ${G\rightarrow H}$ for finitely generated groups ${G}$ and ${H}$. A subgroup is coarsely embedded, hence this is a coarse geometric analogue of studying subgroups.

Example. Although ${{\mathbb Z}^2}$ 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 ${BS(1,2)}$ into a hyperbolic group?

Example. Does there exist an infinite sequence ${G_i}$ such that ${G_i}$ has a coarse embedding into ${G_j}$ iff ${i\leq j}$? Yes, use asymptotic dimension.

Question 2. Does there exist an infinite sequence ${G_i}$ such that ${G_i}$ has a coarse embedding into ${G_j}$ iff ${i\geq j}$? Not so obvious, due to lack of invariants.

1. Local decompositions

Let ${\Gamma}$ be a finite graph. Cutting it into 2 pieces leads to Cheeger’s constant. The ${L^p}$-Poincare constant is an analytic version of Cheeger’s constant.

$\displaystyle \begin{array}{rcl} h^p(\Gamma)=\inf\{\frac{\|\nabla f\|_p}{\|f\|_p}\,;\,\sum_{\mathrm{vertices}}f=0\}. \end{array}$

For ${p=1}$, ${h^1}$ equals Cheeger’s constant up to a factor of 2.

2. Asymptotic decompositions

Definition 1 The ${L^p}$-Poincare profile of an infinite graph ${X}$ is

$\displaystyle \begin{array}{rcl} \Lambda_X^p(n)=\max\{|\Gamma|h^p(\Gamma)\,;\,\Gamma\textrm{ subgraph},\,|\Gamma|\leq n\}. \end{array}$

The joyful stuff is that ${L^p}$-Poincare profile is monotone under coarse embeddings of infinite graphs with bounded degrees.

2.1. What is being measured?

Theorem 2 (Hume 2015) If ${p=1}$, ${\Lambda_X^1}$ coincides with Benjamini-Schramm-Timar’s separation profile.

Proposition 3 (Hume-MacKay-Tessera) If ${p=\infty}$, ${\Lambda_X^\infty}$ depends on volume growth only,

$\displaystyle \Lambda_X^\infty(n)\simeq\frac{n}{\kappa(n)}$

where

$\displaystyle \begin{array}{rcl} \kappa(n)=\min\{k\,;\,\exists c\textrm{ such that }|B(x,k)|\geq n\}. \end{array}$

Since ${\Lambda_X^p}$ increases with ${p}$, we see that ${\Lambda_X^p}$ interpolates between connectivity and growth.

2.2. Examples

1. For a 4-regular tree, ${\Lambda_X^1}$ is bounded, ${\Lambda_X^p(n)\simeq n^{(p-1)/p}}$, ${\Lambda_X^\infty\simeq n/\log n}$.

It follows that ${\Lambda_X^1}$ detects trees.

Proposition 4 (Hume-MacKay) ${\Lambda_X^1}$ is bounded iff ${X}$ is qi to a tree.

2. ${X}$ contains an expander iff ${\Lambda_X^p(n)/n}$ does not tend to 0, for any finite ${p}$. So ${\Lambda_X^p}$ detects expanders.

3. Let ${X={\mathbb Z}^2}$. Then ${\Lambda_X^1(n)\simeq\Lambda_X^\infty(n)\simeq n^{1/2}}$.

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

$\displaystyle \begin{array}{rcl} \Lambda_X^p(n)\leq\frac{n}{\kappa(n)} \end{array}$

for all ${p}$.

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

Question. What about ${X={\mathbb Z}_2\wr{\mathbb Z}^2}$ ? This has (nonlinearly controlled) asymptotic dimension 2 but not finite linearly controlled asymptotic dimension (Nowak).

4. ${\Lambda_X^p}$ detects polynomial growth. Indeed, ${X}$ has polynomial growth iff there exists ${\alpha<1}$ such that for all finite ${p}$, ${\Lambda_X^p(n)\leq n^\alpha}$.

Theorem 6 (Hume-MacKay-Tessera) If ${G}$ is virtually nilpotent with polynomial growth of degree ${d}$, then

$\displaystyle \begin{array}{rcl} \Lambda_X^p(n)\simeq n^{(d-1)/d} \end{array}$

for all ${p}$.

Corollary 7 Rank 1 symmetric spaces satisfy ${\Lambda_X^p(n)\geq n^{(Q-1)/Q}}$ where ${Q}$ is the conformal dimension.

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

Theorem 8 For ${d}$-dimensional real hyperbolic space,

$\displaystyle \begin{array}{rcl} \Lambda_X^p(n)&\simeq& n^{(d-2)/(d-1)}\quad \textrm{ if }pd-1. \end{array}$

Using the Bonk-Schramm embedding, it follows that every hyperbolic graph has polynomial ${\Lambda_X^p(n)\leq n^{(Q-1)/Q}}$ for some ${Q}$, for ${p}$ large.

Question. What is the optimal ${Q}$?

Theorem 9 For Bourdon buildings ${X_{a,b}}$ of conformal dimension ${Q}$,

$\displaystyle \begin{array}{rcl} \Lambda_X^p(n)&\simeq& n^{(Q-1)/Q}\quad \textrm{ if }pQ. \end{array}$

So we clearly see how ${\Lambda_X^p}$ 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 ${X={\mathbb Z}_2\wr{\mathbb Z}}$, for which

$\displaystyle \Lambda_X^1(n)\simeq \frac{n}{\log n}.$

The critical subgraphs are ${\Gamma_k}$ where lit lamps and the lamplighter are in ${[-k,k]}$. The optimal partition of ${\Gamma_k}$ is ${A_k}$

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

However, there is a coarse embedding ${{\mathbb Z}_2\wr{\mathbb Z}}$ into ${BS(1,2)}$. Indeed, both are horospheres in products of trees, for which a Busemann function preserving coarse embedding exists. This answers Question 1.