On the metabelianity of the canonical quotient groups of orientation-preserving line homeomorphisms

For groups G_+ (R) of orientation-preserving line homeomorphisms with a nonempty minimal set a new criterion is obtained for the existence of a projectively invariant Borel measure that are finite on compact sets. It is shown that the existence of a projectively invariant Borel measure finite on compact sets is equivalent to the metabelianity of the canonical quotient group G/HG, where the normal subgroup HG consists of the homeomorphisms in G that fix all points in the minimal set. It is shown that for groups G_+ (R) of orientation-preserving line homeomorphisms with a nonempty minimal set, in the space of quotient groups G/HG the class of metabelian groups coincides with the class of groups with finite normal series whose quotients contain no free subsemigroups with two generators, and the class of abelian groups coincides with the class of groups not containing free subsemigroups with two generators. On this basis, for the class of solvable groups G_+ (R) of orientation-preserving line homeomorphisms with a nonempty minimal set it is proved that the group G/HG is metabelian. For the original group G_+ (R) of orientation-preserving line homeomorphisms with a nonempty minimal set and a nontrivial quotient group which does not contain a freely acting homeomorphism it is shown that it is combinatorially complex: such a group is not a group with finite normal series the quotients of whose terms contain no free subsemigroups with two generators. Bibliography: 16 titles.