A condition that prevents groups from acting non-trivially on trees

Martin R. Bridson

Preprint, Oct 2006 (edited June 2007): temporarily removed for a technical reason; will return shortly

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 triple of 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.