Variants of the Bonk-Schramm embedding

Question

Recently I heard about the following embedding theorem of Bonk and Schramm: every Gromov hyperbolic geodesic metric space with "bounded growth" is roughly similar to a convex subset of $\mathbb{H}^k$ for some integer $k$. I find this result very interesting (Nash embedding theorem has always been a personal favourite), and I am very curious to know if there are variants of this result in literature. For example, one can think of the following variants:

1. If we replace $\mathbb{H}^k$ by a Hadamard manifold with variable (but bounded) curvature, can we embed more general spaces? In particular, I am not sure about the role of "bounded growth" in the Bonk-Schramm embedding. Can this requirement be dropped?

2. If we relax the requirement on the map $f$ itself, and allow it to be a quasi-isometry, or even a coarse embedding, can we expect more general spaces on the two ends?

I am a novice in this area, and the literature seems to be huge. So any help is deeply appreciated, and I apologize if my questions are trivial.

Answer 1

Your question (actually, questions) is a bit too vague for my taste. Here is an answer of sorts.

1. If we replace $\mathbb{H}^k$ by a Hadamard manifold with variable (but bounded) curvature, can we embed more general spaces?

The answer will depend on which Hadamard manifolds you are willing to use. For concreteness, if you allow ${\mathbb H}^n\times {\mathbb R}$ then obviously you can embed more general metric spaces, e.g. the Euclidean plane. You can go even further and allow symmetric spaces of noncompact type as your targets (where the rank is not a priori bounded). This will allow you to embed even more spaces. Etc.

In particular, I am not sure about the role of "bounded growth" in the Bonk-Schramm embedding. Can this requirement be dropped?

No, it cannot be dropped since bounded growth is preserved under quasi-isometric embeddings. For instance, if your metric space $T$ is a tree of locally finite but unbounded valence, you cannot quasi-isometrically embed $T$ in any Hadamard manifold of curvature bounded below.

2. If we relax the requirement on the map $f$ itself, and allow it to be a quasi-isometry can we expect more general spaces?

I assume, you are still trying to embed in finite-dimensional real-hyperbolic spaces. No, you will get the same class of geodesic metric spaces, namely, Gromov-hyperbolic of bounded growth. The reason is that bounded growth and hyperbolicity are preserved under quasi-isometric embeddings and convex subsets of ${\mathbb H}^n$ (with the induced metric) are Gromov-hyperbolic and of bounded growth.

If we relax the requirement on the map $f$ itself, and allow it to be a coarse embedding can we expect more general spaces?

Yes, of course. For instance, horospheres in ${\mathbb H}^n$ (equipped with the intrinsic path-metrics) are coarsely embedded copies of the Euclidean space $E^{n-1}$. Similarly, one can coarsely embed every finitely generated nilpotent group in some ${\mathbb H}^n$.

Thus, you can embed spaces which are not Gromov-hyperbolic. But you still have some restrictions, such as bounded growth. On the other hand, there are finitely generated groups which cannot be coarsely embedded in ${\mathbb H}^n$ (say, solvable groups of exponential growth, see the paper by Hume and Sisto "Groups with no coarse embeddings into hyperbolic groups"). Thus, there are obstructions besides bounded growth.