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

For groups $G\subseteq\operatorname{Homeo}+({\mathbb R})$ of orientation-preserving line homeomorphisms with a nonempty minimal set a new criterion is obtained for the existence of a projectively invariant Borel measure 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 $G\subseteq\operatorname{Homeo}+({\mathbb R})$ 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 $G\subseteq\operatorname{Homeo}+({\mathbb R})$ of orientation-preserving line homeomorphisms with a nonempty minimal set, a nontrivial quotient group and without 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.

groups of line homeomorphisms, canonical quotient groups, metabelianity