A group $G$ has Serre's property $FA$ if any isometric action of $G$ on a simplicial tree has a global fixed point. Let $n\geq 3$. It is well-known that $SL\_n(\mathbb{Z} )$ has property $FA$. Now my question is that are there nontrivial group actions of $SL\_n(\mathbb{Z} )$ on a simplicial tree by isometries? Here "nontrivial" means the fixed point set is NOT the whole tree. 



Non-trivial action of $SL_n(\mathbb{Z} )$ on a simplicial tree
Any non-trivial group $G$ has a non-trivial action on a "star" tree $T$ whose vertex set is $G\cup\{\infty\}$ (where $\infty\notin G$) and edges are $\{\infty,g\}$ for $g\in G$. Thus any group admits a faithful action on a tree.


Any residually finite countable group has a faithful action on a *locally finite* tree. If $(H\_n)$ is a decreasing sequence of finite index subgroups with trivial intersection with $H\_0=G$, this tree is the disjoint union of cosets $G/H\_n$, with an edge between $gH\_n$ and $gH\_{n+1}$ for all $g$ and all $n$; the root is just the point $G/H\_0$ and is fixed by the action. This applies to $\text{SL}\_d(\mathbf{Z})$; in this precise case the $H\_n$ can be chosen so that the index $H\_n/H\_{n+1}$ is bounded and thus the tree has bounded valency.


