Left ordered amenable and locally indicable groups

Peter A. Linnell

There has been interest recently concerning when a left ordered group is locally indicable. Chiswell and Kropholler proved that every left ordered solvable-by-finite group is locally indicable, while Bergman gave examples of left ordered groups which are not locally indicable. In this paper it is proved that every left ordered elementary amenable group is locally indicable. Recall that the class of elementary amenable groups is the smallest class of groups which contains all abelian-by-finite groups, is closed under group extension, and is closed under directed unions. It is well known that every solvable-by-finite group is elementary amenable, and every elementary amenable group is amenable. It is left as an open problem to as whether every left ordered amenable group is locally indicable.

The key idea in this paper is to view a left ordered group G as orientation preserving homeomorphisms of $\mathbb {R}$ . One then has a sort of Galois theory between subgroups of G and closed subsets of $\mathbb {R}$ . This is done by using the fixed points of a subgroup to be the corresponding closed subset of $\mathbb {R}$ , and by using the pointwise stabilizer of a closed subset of $\mathbb {R}$ to be the corresponding subgroup of G.