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 }p<d-1,\\ &\simeq& n^{(p-1)/p}\log(n)^{1/p}\quad \textrm{ if }p=d-1,\\ &\simeq& n^{(p-1)/p}\quad \textrm{ if }p>d-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 }p<Q,\\ &\simeq& n^{(p-1)/p}\log(n)^{1/p}\quad \textrm{ if }p=Q,\\ &\simeq& n^{(p-1)/p}\quad \textrm{ if }p>Q. \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.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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