I have the following question regarding group actions on trees to which I suspect the answer to be "yes", but it could very well be that extra conditions are required (it is certainly true for free actions on trees):

Let $G$ be a finitely generated group acting by simplicial automorphisms on a minimal simplicial tree $T$ (here, *minimal* means that $T$ does not have a $G$-invariant proper subgraph, and it implies that $T$ is cocompact). We call $g\in G$ *elliptic* if it fixes a point in $T$ and *hyperbolic* if not.

Moreover, let $R\subset T$ be an infinite ray based at a vertex $x_0\in V(T)$. Since $T$ is cocompact and therefore has only finitely many orbits of vertices, when walking along $R$ we eventually find a vertex $y\in V(R)$ that is a $G$-translate of some 'earlier' vertex, i.e. $y=gy_0$ for some $y_0\in [x_0,y)\subset R$ and $g\in G$. However, we do not know whether $g$ is elliptic or hyperbolic.

**Question:** Does there exist $y\in V(R)$ such that $y=gy_0$ for some $y_0\in [x_0,y)\subset R$ and some *hyperbolic* $g\in G$?

I ended up considering two cases and I have sorted out one of the two. The other one, however, seems to break up into many other cases again that I cannot yet handle. I decided not to post my partial result in order not to lead people in the (possibly) wrong direction from the very beginning.