Coarse embeddings and Gromov products in (Gromov) hyperbolic spaces

I am new into geometric group theory and I have recently started reading the book "Sur les Groupes Hyperboliques d’après Mikhael Gromov" by Ghys and de la Harpe. The following inequality regarding Gromov products and quasi-isometries on a hyperbolic metric space $$X$$ caught my attention:

For a $$C$$-quasi-isometric map $$f$$, $$C^{-1} (x,y)_z-A \leq (f(x), f(y))_{f(z)} \leq C(x,y)_z +A$$ for all $$x,y,z \in X$$, where $$A$$ is a constant depending on the quasi-isometry constants of $$f$$ and the hyperbolicity constant of $$X$$.

The natural question that came to my mind is whether an analogous inequality might hold if $$f$$ is just a coarse embedding. It seems that hyperbolicity may not be preserved by coarse embeddings so my guess is that a nice relation like above may not be expected, but can we expect any relation between the Gromov products $$(x,y)_z$$ and $$(f(x), f(y))_{f(z)}$$?

I really appreciate any help. Thanks in advance!

• What are your actual assumptions when you say that "$f$ is just a coarse embedding?" Do you mean a coarse embedding of general metric spaces? From one hyperbolic space to another? Do you assume that hyperbolic spaces are geodesic? ... Commented Mar 25 at 11:29
• @MoisheKohan (1) By a coarse embedding $f$ between metric spaces $X$ and $Y$ we mean a map satisfying $\phi_-(d(x,y))\leq d(f(x), f(y)) \leq \phi_+(d(x,y))$ for all $x,y\in X$, where $\phi_-, \phi_+$ are non-decreasing, non-negative functions such that $\lim _{t\rightarrow\infty} \phi_-(t)=\infty$. If it helps, you can also assume $X$ and $Y$ to be hyperbolic. (2) Yes we assume a hyperbolic space to be geodesic. Commented Mar 25 at 12:10
• If you put $x=y$ in your inequalities, you deduce that $f$ has to be a quasi-isometric embedding. Commented Mar 25 at 12:22
• For maps of hyperbolic spaces your question is related to Cannon-Thurston maps. In general, even for coarse embeddings of hyperbolic spaces, you will get neither upper nor lower bounds. Commented Mar 25 at 12:23

Even for coarse maps between Gromov-hyperbolic spaces $$f: X\to Y$$ there are neither reasonable upper nor lower bounds of the type $$\psi_-((x,y)_z)\le (f(x), f(y))_{f(z)}\le \psi_+((x,y)_z)$$ (where $$\psi_-$$ is proper and $$\psi_+$$ is continuous). Namely, you should expect to have sequences $$(x_n), (x'_n)$$ in $$X$$ such that:
1. $$(x_n,x'_n)_z$$ is uniformly bounded while $$(f(x_n), f(x'_n))_{f(z)}$$ is unbounded.
This happens, for instance, in the situation when the coarse map $$f: X\to Y$$ has a continuous non-injective extension to a map (called a Cannon-Thurston map) $$\partial X\to \partial Y$$ of Gromov-boundaries of $$X, Y$$. Examples like this abound. For instance, consider the inclusion map of a horocycle $$X$$ (with the induced path-metric) in the hyperbolic plane $$Y$$.
1. $$(x_n,x'_n)_z$$ diverges to infinity while $$(f(x_n), f(x'_n))_{f(z)}$$ is uniformly bounded. This happens when the map $$f$$ does not have a Cannon-Thurston extension. Examples like this exist but are harder to construct, especially in the group-theoretic setting.