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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!

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  • $\begingroup$ 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? ... $\endgroup$
    – Moishe Kohan
    Commented Mar 25 at 11:29
  • $\begingroup$ @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. $\endgroup$
    – Steve
    Commented Mar 25 at 12:10
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    $\begingroup$ If you put $x=y$ in your inequalities, you deduce that $f$ has to be a quasi-isometric embedding. $\endgroup$
    – AGenevois
    Commented Mar 25 at 12:22
  • $\begingroup$ 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. $\endgroup$
    – Moishe Kohan
    Commented Mar 25 at 12:23

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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.

There are several papers by Mahan Mj on Cannon-Thurston maps, see for instance, the survey

Mj, Mahan, Cannon-Thurston maps, Sirakov, Boyan (ed.) et al., Proceedings of the international congress of mathematicians, ICM 2018, Rio de Janeiro, Brazil, August 1–9, 2018. Volume II. Invited lectures. Hackensack, NJ: World Scientific; Rio de Janeiro: Sociedade Brasileira de Matemática (SBM). 885-917 (2018). ZBL1447.57027.

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