# Group acting on a CAT(0) cube complex then acting also on a tree

If a group $G$ acts on a CAT(0) cube complex, then does $G$ act on a simplicial tree?

The most elementary example seems to be the triangle group $$T= \langle a,b,c \mid a^2=b^2=c^2=(ab)^3=(ac)^3=(bc)^3=1 \rangle.$$ It satisfies Serre's property (FA) since it is generated by torsion elements such that the product of any two of these elements has finite order. On the other hand, $T$ is the symmetry group of the tesselation of the plane by equilateral triangles. It naturally defines a collection of walls on the plane, so that by cubulating this wallspace, you deduce that $T$ acts properly on $\mathbb{R}^3$ (though of as a cube complex by looking at its usual tesselation with cubes).