Can – or Can’t – act on Trees.

Quick overview on Bass-Serre Theory:

One of crucial topics in Geometric Group Theory is Bass-Serre theory. In this theory actions on trees is used to understand a group. One might asks why only trees. Because the combinatorics and geometry of trees is very nice: for instance, there is a geodetic(unique path) between any two vertices or there is no cycle or loop. When a group acts on a tree, we can associate a very important structure called graph of groups to it. This structure can be described as  a graph labelled by a collection of subgroups (the stabilisers) to the action. Two of toy examples with one single edge is the notions of an amalgamated free product and an HNN extension. More complicated examples can be obtained by applying these constructions one edge at a time. In an amalgamated free product there are two groups G1  and G2 , with isomorphic subgroups H1 and H2 . We glue G1 and G2 together by identifying an element of H1 with the corresponding element of H2. For example in the following figure the vertex u corresponded to G1 and the vertex v corresponded to G2 and the edge uv corresponds to H1\cong H2.

For an HNN extension, we have the group G, and moreover two isomorphic subgroups H1 and H2 of G. Then we add a new element t (corresponding to the loop), and declares that conjugating an element of H1 by t gives the corresponding element in H2. In the following figure, u corresponds to G and e corresponds to H1\cong H2

Serre’s Property (FA)

It's reasonable if we expect all groups act on trees in a not trivial way. For instance, if we consider the class of all finite trees or rooted trees, then all actions must have some sort of fixed point(vertex or edge) and it doesn't provide the kind of structure we’re looking for. Thanks to Jean-Peirre Serre for introducing the property (FA). A group G is said to have property FA if every action of G on a tree has a fixed point. Property FA is equivalent for countable G to the three properties:

• Not splitting as an amalgamated free product

• Admitting no quotient isomorphic to Z, and

• Being finitely generated

The following groups have property FA: a finitely generated torsion group or SL3(Z), however SL2(Z) doesn't have property FA.