Examples of hyperbolic and non-hyperbolic space for quasi-isometric spaces

Question

Let $X$ and $Y$ are quasi-isometric spaces. I try to find an example for which one of these spaces will be hyperbolic, other is not hyperbolic.

I know that for geodesic metric space if one of the quasi-isometric spaces is a hyperbolic space also it's truth for second space.

It's interesting to see some examples where the hyperbolicity is not a quasi-isometric invariant.

Thank you!

Answer 1

Here is the "standard example:"

Let $X$ denote the real line with the standard metric. Let $Y\subset {\mathbb R}^2$ be the graph of the function $f(x)=|x|$ with the metric (the distance function) obtained by restriction of the standard metric on the plane. I will leave it to you to check that $X$ is hyperbolic, $Y$ is not (in the sense of the definition via the Gromov-product) and to construct a quasi-isometry $X\to Y$.