Property FA is a property of mathematical group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

s. Jean-Pierre Serre

Jean-Pierre Serre

Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...

 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

Group action

In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...

 of G on a tree

Tree (graph theory)

In mathematics, more specifically graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one simple path. In other words, any connected graph without cycles is a tree...

 has a global fixed point

Fixed point (mathematics)

In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...

.

Serre shows in Trees that if a group has property FA, then it cannot split as a free product with amalgamation

Free product

In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “most general” group having these properties...

 or HNN extension

HNN extension

In mathematics, the HNN extension is a basic construction of combinatorial group theory.Introduced in a 1949 paper Embedding Theorems for Groups by Graham Higman, B. H...

. In particular, a finitely generated

Generating set of a group

In abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their...

 group with property FA has finite abelianization.

It is a theorem of Watatani that Kazhdan's property (T)

Kazhdan's property (T)

In mathematics, a locally compact topological group G has property if the trivial representation is an isolated point in its unitary dual equipped with the Fell topology. Informally, this means that if G acts unitarily on a Hilbert space and has "almost invariant vectors", then it has a nonzero...

implies property FA of Serre.