I have these two questions that I cannot get any intuition about. Perhaps someone can possibly offer a few hints on how to get started?
1) Show that the ends of ${\bf F_2} \oplus {\bf F_2}$ is equal to 1 (i.e, $e({\bf F_2} \oplus {\bf F_2}) =1$). The reason this is confusing me is because $e({\bf F_2}) = \infty$, so I do not know how my question is equal to 1.
2) Let $G$ be a finitely generated group with $e(G)=2$ and $\lambda$ be its cayley graph. There exists a finite subgraph $C$ such that $\lambda$ \ $C$ has exactly two connected, unbounded components. We choose one of these two complements, and now let $E \subset G$ consist of the elements of $G$ that correspond to the vertices in a connected component that is unbounded. Note that $g \in G$. I need to show that: either $E \triangle gE$ is finite or $(E \triangle gE)^{c}$ is finite.
Any help would be great.