So I'm reading Bowditch's lecture notes on geometric group theory and gave this as an example, but I didn't quite follow his reasoning. I was wondering if someone could explain why these two spaces are not quasi isometric. Thanks.
Excerpt from Bowditch:
EDIT: So I get everything up to the point where he derives a contradiction. I don't see why $|\phi(0)-\phi(p)|$ agrees with $a-\phi(0)$ up to a constant where the constant is independent of $a$. This is what is troubling me.