The following questions might be trivial, however, I couldn't solve them:
Let $G$ be generated by a finite symmetric set $S.$ Suppose that $\Gamma(G,S)$ is the corresponding right Cayley graph of $G.$ $X$ is a metric space(or, maybe a topological space with some nice structure).
(1) Is there a way to check the following property of a space $X:$ $X$ is not quasi-isometric to a space $Z$ which is quasi-isometric to a(hence, every) Carley graph $\Gamma(G,S)$ of some f.g group $G.$
I.e, If we partition the space of spaces upto quasi-isometric equivalence then does every equivalence class contain a space which is quasi-isometric to a Cayley graph of some f.g group $G?$
(2) By Stalling's theorem, # of ends is a geometric property of the group. Does this mean that # of ends is a quasi-isometric invariant of the spaces which are quasi-isometric to Cayley graphs?
If the answer of question #2 is affirmative and equivalence class question above fails; i.e, there is an equivalence class whose elements are not quasi-iso. to any Cayley graph ; then what is the example of spaces $W_1, W_2$ which are not quasi-iso. to any Cayley graph but $W_1$ is quasi-isometric to $W_2$ ,however, # of ends of $W_1$ is different than the # of ends of $W_2.$