The classification theorem for groups of homeomorphisms of the line.   Nonamenability of Thompson's group $F$
  This paper allows one to obtain a criterion for the existence of a projectively invariant measure formulated in terms of combinatorial properties of a group (amenability of some canonical quotient group). Such necessary and sufficient condition is a basis of the classification scheme for groups of homeomorphisms of the line. In particular, a nonamenability of Thompson's group $F$ follows from the obtained criterion. 
Leva Beklaryan
math.GR math.DS