Serre's property FA explained

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 SL₃(Z) and more generally G(Z) where G is a simply-connected simple Chevalley group of rank at least 2. The group SL₂(Z) is an exception, since it is isomorphic to the amalgamated product of the cyclic groups C₄ and C₆ along C₂.

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.

Examples

The following groups have property FA:

• A finitely generated torsion group;
• SL₃(Z);
• The Schwarz group

\left\langle{ a,b : a ^{A} = b ^{B} = (ab) ^{C} = 1 }\right\rangle for integers A,B,C ≥ 2;

• SL₂(R) where R is the ring of integers of an algebraic number field which is not Q or an imaginary quadratic field.

The following groups do not have property FA:

• SL₂(Z);
• SL₂(R_D) where R_D is the ring of integers of an imaginary quadratic field of discriminant not −3 or −4.

References

• Book: Serre, Jean-Pierre . Jean-Pierre Serre

. Proceedings of the Second International Conference on the Theory of Groups . Amalgames et points fixes . French . Jean-Pierre Serre . Lecture Notes in Mathematics . 372 . 1974 . 633–640 . 0308.20026 . 0376882 .

• Book: Serre, Jean-Pierre . Jean-Pierre Serre

. Arbres, amalgames, SL₂ . French . Jean-Pierre Serre . Astérisque . 46 . Société Mathématique de France . 1977 . 0369.20013 . English translation: Book: Serre, Jean-Pierre . Jean-Pierre Serre

. Trees . Jean-Pierre Serre . Springer . 2003 . 3-540-44237-5 . 1013.20001 .

• Yasuo . Watatani . Property T of Kazhdan implies property FA of Serre. . Math. Japon. . 27 . 97–103 . 1981 . 0489.20022 . 649023 .