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 1 The -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,
Since increases with , we see that interpolates between connectivity and growth.
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 7 Rank 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 8 For -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 9 For 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.