Groups of homeomorphisms of the line. Criteria for the existence of invariant and projectively invariant measures in terms of the commutator subgroup

Existence criteria for invariant and projectively invariant measures are obtained for a group $G$ of homeomorphisms of the line. These criteria are formulated in terms of the commutator subgroup $[G,G]$. For the special (but very important) case of groups of homeomorphisms of the line containing a freely acting element we obtain a criterion for the existence of a projectively invariant measure in the form of the absence of a special subgroup with two generators in which one of the generating elements is a freely acting element.
Bibliography: 20 titles.