# Serre's property FA

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 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.

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