Good time of day!
I'm trying to show that: "The Cayley graph of a finitely generated group $G$ is quasi-isometric to a line if and only if $G$ has a cyclic subgroup of finite index".
My thoughts about this: I was trying to apply Svarc-Milnor lemma which states
" Let G be a group acting by isometries on a proper length space $X$ such that the action is properly discontinuous and cocompact. Then the group $G$ is finitely generated and for every finite generating set $S$ of $G$ and every point $p\in X$ the orbit map
$f_{p}:(G,d_{S}) \to X$, $g \to gp$ is a quasi-isometry. Here ${\displaystyle d_{S}}$ is the word metric on $G$ corresponding to $S$.
I'm not sure about this attempt. Please can you explain this in more details? Thank you!