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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 $⟨a,b:a^A=b^B=(ab{)}^C=1⟩$ 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

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