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Amenable groups that act on the line

Dave Witte Morris

Algebraic & Geometric Topology 6 (2006) 2509–2518

arXiv: math.GR/0606232


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.

amenable, action on the line, action on the circle, ordered group, indicable, cyclic quotient
Mathematical Subject Classification 2000
Primary: 20F60
Secondary: 06F15, 37C85, 37E05, 37E10, 43A07, 57S25
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Received: 9 June 2006
Accepted: 1 September 2006
Published: 15 December 2006
Dave Witte Morris
Department of Mathematics and Computer Science
University of Lethbridge
Alberta, T1K 3M4