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 $G$. Let $X$ be a metric space (or, maybe, a topological space with some nice structure).
(1) Is there a way to check the following property: $X$ 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 up to 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, the number of ends is a geometric property of the group. Does this mean that the number 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 if the question (1) about equivalence classes has a negative answer, i.e., there is an equivalence class whose elements are not quasi-isomorphic to any Cayley graph, then what is an example of spaces $W_1, W_2$ which are not quasi-isometric to any Cayley graph, but $W_1$ is quasi-isometric to $W_2$ ,however, the number of ends of $W_1$ is different from the number of ends of $W_2$?