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 $W\_2$? 



Quasi-isometries vs Cayley Graphs
I guess that a star (a tree with $n$ infinite branches issued from a single vertex) should answer at least your first question. It should have $n$ ends, whatever meaningful definition you use, an we know that a group has $1$, $2$ or an infinity of ends.


Since quasi-isometry is an equivalence relation, you do not need to invoke a space $Z$ in your first question and the answer of your second question is obviously positive.


