Serre's property FA
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Property FA is a property of mathematical groups. Jean-Pierre Serre defined property FA in his book Arbres, amalgames, 
(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 has 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.
