Serre's property FA

Property FA is a property of mathematical groups. Jean-Pierre Serre defined property FA in his book Arbres, amalgames, $SL_{2}$' (published in English, translated by John Stilwell, as Trees).

A group G is said to have property FA if every action of G on a tree has a global fixed point.

Serre shows in Trees that if a group property FA, then it cannot split as a free product with amalgamation or HNN extension. In particular, a finitely generated group with property FA has finite abelianization.

It is a theorem of Watatani that Kazhdan's property (T) implies property FA of Serre.