R and R^2 not quasi-isometric?

[nonequilibrium]

Hello,

I'm reading a book on geometric group theory (and because this can be considered to be a part of multiple subsets of math, I chose to post it in the general math forum; correct me if this was incorrect).

One of the exercises in the book is "show that $\mathbb R$ and $\mathbb R^2$ are not quasi-isometric". This was shortly after the definition of quasi-isometry, so there's no real arsenal for it, so I suppose one has to prove it directly from definition (from contraposition?), but I don't really see it happening. Can anybody help me out?

EDIT: Sorry! Wrong forum... I just read the "MUST READ" and apparently questions like this should go into the homework questions forum, my apologies! Can somebody move it?

 

[mighty2000]

Hello,

Thank you for sharing your question about geometric group theory. It is great to see that you are actively working through exercises in your book and seeking help when needed.

To answer your question, let me first explain the concept of quasi-isometry. A quasi-isometry is a function between two metric spaces that preserves distances up to a constant factor. In other words, if two points are close together in one space, their images will also be close together in the other space, and vice versa. This means that quasi-isometric spaces have similar geometric properties, even though they may look different.

Now, to prove that \mathbb R and \mathbb R^2 are not quasi-isometric, we need to show that there is no function that satisfies the definition of a quasi-isometry between these two spaces. One way to do this is by using the contrapositive statement of the definition. If we assume that there exists a quasi-isometry f: \mathbb R \to \mathbb R^2, then we can show that this leads to a contradiction.

For example, let's consider the points (0,0) and (1,0) in \mathbb R^2. Since f is a quasi-isometry, there should exist a constant C such that the distance between these two points is preserved. However, in \mathbb R, the distance between the images of these points would be \sqrt{2}, which is not equal to the distance between (0,0) and (1,0).

This is just one possible approach to prove that \mathbb R and \mathbb R^2 are not quasi-isometric. I encourage you to think about other ways to prove it as well. Also, keep in mind that this is just one exercise and there may be more to come that will deepen your understanding of quasi-isometries.

I hope this helps! Good luck with your studies and feel free to ask more questions if needed.