Algebraic & Geometric Topology 6 (2006) 2509–2518

DOI: 10.2140/agt.2006.6.2509

arXiv: math.GR/0606232

Abstract

Let Γ be a finitely generated, amenable group. Using an idea of É Ghys, 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.

Keywords

amenable, action on the line, action on the circle, ordered group, indicable, cyclic quotient

Mathematical Subject Classification

Primary: 20F60

Secondary: 06F15, 37C85, 37E05, 37E10, 43A07, 57S25

References

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Publication

Received: 9 June 2006
Accepted: 1 September 2006
Published: 15 December 2006

Authors