I am trying to prove that every two ended finitely generated group is virtually $\mathbb{Z}$. My first idea is to find an element $g\in G$ such that $\mathbb{Z} = <g>$.
$e(G) = 2$, so there is a compact subset $K$ of Cayley graph of $G$, $\Gamma$, such as $1\in K$, and $\Gamma\setminus K$ has two unbounded components $\Gamma_1, \Gamma_2$. Let $g\in G$ and $|g| > 2\cdot\hbox{diam}(K)$. WLoG $g\in\Gamma_1$. Now I want to show, that
$\Gamma_1\supset g\Gamma_1\supset g^2\Gamma_1\supset\ldots$.
I tried to show this for every $|g|> 2\cdot\hbox{diam}(K)$, but obviously this is not true. How to choose $g$?
Maybe I need to consider group action on $Ends(G)$ but I don't know how to define it.