Let $G$ be a finitely generated group. Fix some symmetric finite generating set $S$ for $G$, and write $\Gamma$ for the Cayley graph of $G$ with respect to $S$.

Given finite subsets $X,S,Y$ of $G$, we say that $S$ separates $X$ from $Y$, denoted $(X:S:Y)$, if $X$ and $Y$ lie in distinct connected components of $\Gamma\setminus S$. If $B_n$ denotes the $n$-ball in $G$, we say that $g\in G$ is $n$-axial if

$(g^a B_n: g^b B_n : g^c B_n)$

whenever $a<b<c$ are integers. Hopefully, it is easy to imagine what such elements look like in $\mathbb{Z}$ or $\mathbb{Z}\ast\mathbb{Z}$, where some power of any nontrivial element will be $n$-axial. On the other hand, there are no $n$-axial elements in a one ended group like $\mathbb{Z}^2$.

My PhD thesis claims that, when $G$ has at least two ends, $n$-axial elements exist for $n$ sufficiently large. Sadly, Ville Salo and Ilkka Törmä recently pointed out to me that my proof is wrong.

Before returning my degree, it seems prudent to ask:

If $G$ has at least two ends, must $G$ contain an $n$-axial element for some $n$?

I also wonder about the Bass-Serre theory interpretation of this problem. That is, suppose that $G$ acts nontrivially (without global fixed point) with finite edge stabilizers on a tree $T$. If $g\in G$ acts hyperbolically on $T$, must some power of $g$ be $n$-axial?