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.
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.
If some subgroup of finite index in G has property FA then so does G, but the converse does not hold in general.
Indeed, any subgroup of finite index in a T-group has property FA.