Solvable Groups Acting on the Line

Actions of discrete groups on the real line are considered. When the group of homeomorphisms is solvable several sufficient conditions are given which guarantee that the group is semiconjugate to a subgroup of the affine group of the line. In the process of obtaining these results sufficient conditions are also determined for the existence of invariant (quasi-invariant) measures for abelian (solvable) groups acting on the line. It turns out, for example, that any solvable group of real analytic diffeomorphisms or a polycyclic group of homeomorphisms has a quasi-invariant measure, and that any abelian group of C diffeomorphisms has an invariant measure. An example is given to show that some restrictions are necessary in order to obtain such conclusions. Some applications to the study of codimension one foliations are indicated. Introduction. If G is a group of homeomorphisms of the line it is reasonable to ask for a dynamic description of how G acts when reasonable restrictions are placed on G. For example, in [9] it is shown that if G is finitely generated and nilpotent then there is a G-invariant Borel measure on R which is finite on compact sets. This already says much about G. The proof is quite different from classical arguments which guarantee the existence of a finite invariant measure when an amenable group acts on a compact Hausdorff space [6] in that one uses pseudogroup properties of G rather than group properties (specifically, nonexponential growth of orbits rather than nilpotence of G). It is reasonable to determine what can be said when G is solvable, but, as pointed out in [9], solvable groups need not have invariant measures, e.g., subgroups of the affine group which contain nontrivial translations and nontrivial dilations. Such examples suggest consideration of a more general invariance property. A measure is called quasi-invariant (for G) if each element of G multiplies the measure by a nonzero constant which depends on the particular element of G. Usual Lebesgue measure is quasi-invariant for the affine group so it is reasonable to ask if every (finitely generated) solvable G has a quasi-invariant measure. It turns out that a large class of solvable groups (including polycyclic groups) acting on the line must have quasi-invariant measures but there are solvable groups which do not have such measures. An important step in the investigation of solvable groups is the consideration of infinitely generated abelian groups. If differentiability is assumed, stronger Received by the editors October 30, 1980 and, in revised form, August 3, 1982. 1980 Mathematics Subject Classification. Primary 57E25; Secondary 58F11, 58F15, 58F18.