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. 



Group actions on trees and translates under hyperbolic elements
The answer is yes. For the proof, the ray $R$ contains three distinct points in order, $y\_0,y\_1,y\_2$, such that $y\_1 = g\_1 y\_0$ and $y\_2 = g\_2 y\_1$ for some elements $g\_1,g\_2 \in G$. If either of $g\_1,g\_2$ is hyperbolic, you are done. Suppose that both of $g\_1,g\_2$ are elliptic. The midpoint $p\_1$ of $[y\_0,y\_1]$ must be fixed by $g\_1$, and so $g\_1$ takes the segment $[y\_0,p\_1]$ to the segment $[y\_2,p\_1]$. Similarly, $g\_2$ fixes the midpoint $p\_2$ of $[y\_1,y\_2]$ and takes $[y\_1,p\_2]$ to $[y\_2,p\_2]$. Consider $g = g\_2 g\_1$ which takes $y\_0$ to $y\_2$, and note by construction that $g[y\_0,y\_2] \cap [y\_0,y\_2] = \lbrace y\_2 \rbrace$, so $g$ is hyperbolic and $[y\_0,y\_2]$ is a fundamental domain for the axis of $g$.


