In this short note written for the
Zieschangschrift, I describe a simple condition on the
finite subgroups of a group which ensures that the group
will have property FA (in fact FR); I call this the
Triangle Criterion. Aut(F_n) and SL(n,Z)
satisfy this condition if n is at least 3. In summary, the
proof is as follows.
One takes three subgroups of Aut(F_n)
obtained as the pairwise intersections
of the symmetry groups of a certain
graphs. The union of these
three groups generates Aut(F_n), which therefore satisfies
the Triangle Criterion.
The graphs are shown in the figure below. I am grateful to Karen Vogtmann for drawing this figure.