In mathematics, Property FA is a property of groups first defined by Jean-Pierre Serre.
A group G is said to have property FA if every action of G on a tree has a global fixed point.
Serre shows that if a group has property FA, then it cannot split as an amalgamated product or HNN extension; indeed, if G is contained in an amalgamated product then it is contained in one of the factors. In particular, a finitely generated group with property FA has finite abelianization.
Property FA is equivalent for countable G to the three properties: G is not an amalgamated product; G does not have Z as a quotient group; G is finitely generated. For general groups G the third condition may be replaced by requiring that G not be the union of a strictly increasing sequence of subgroup.
Examples of groups with property FA include SL<sub>3</sub>(Z) and more generally G(Z) where G is a simply-connected simple Chevalley group of rank at least 2. The group SL<sub>2</sub>(Z) is an exception, since it is isomorphic to the amalgamated product of the cyclic groups C<sub>4</sub> and C<sub>6</sub> along C<sub>2</sub>.
Any quotient group of a group with property FA has property FA. If some subgroup of finite index in G has property FA then so does G, but the converse does not hold in general. If N is a normal subgroup of G and both N and G/N have property FA, then so does G.
It is a theorem of Watatani that Kazhdan's property (T) implies property FA, but not conversely. Indeed, any subgroup of finite index in a T-group has property FA.
The following groups have property FA:
The following groups do not have property FA: