un 2 00 6 Amenable groups that act on the line

Let Γ be a finitely generated, amenable group. We prove that if Γ has a nontrivial, orientation-preserving action on the real line, then Γ has an infinite, cyclic quotient. (The converse is obvious.) This implies that if Γ has a faithful action on the circle, then some finite-index subgroup of Γ has the property that all of its nontrivial finitely generated subgroups have infinite, cyclic quotients. It also means that every left-orderable, amenable group is locally indicable. This answers a question of P. Linnell. Let Γ be an abstract group (with the discrete topology). It is obvious that if Γ has an infinite cyclic quotient, then Γ has a nontrivial, orientation-preserving action on the real line R. The converse is not true in general, even for finitely generated groups [4, Eg. 6.9.2]. In this note, we prove that the converse does hold in the class of finitely generated amenable groups. • A measure µ on a measure space X is said to be a probability measure iff µ(X) = 1. • A (discrete) group Γ is amenable iff for every continuous action of Γ on a compact, Hausdorff space X , there is a Γ-invariant probability measure on X. Theorem A Let Γ be a finitely generated, amenable group. Then Γ has a nontrivial, orientation-preserving action on R if and only if Γ has an infinite cyclic quotient. 1 It is well known that a countable group has a faithful, orientation-preserving action on R if and only if it is left orderable [3, Thm. 6.8]. (That is, there is a total order ≺ on Γ, such that, for all γ, λ 1 , λ 2 ∈ Γ, if λ 1 ≺ λ 2 , then γλ 1 ≺ γλ 2 .) Also, every subgroup of an amenable group is amenable [10, Prop. 13.3]. Hence, the nontrivial direction of Thm. A can be stated in the following purely algebraic form. Definition 0.2 [4, p. 127] A group is locally indicable iff each of its nontrivial finitely generated subgroups has an infinite cyclic quotient. Theorem B Every amenable left-orderable group is locally indicable. (2) Every locally indicable group (whether amenable or not) is left orderable [1],[4, Lem. 6.9.1]. (3) Theorem B has previously been proved with " amenable " replaced by stronger hypotheses, such as solvable-by-finite [2], supramenable [5], or elementary amenable [6]. There are also interesting results that replace " amenable …