Amenable groups that act on the line

Morris, Dave Witte

doi:10.2140/agt.2006.6.2509

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Amenable groups that act on the line

Dave Witte Morris

Algebraic & Geometric Topology 6 (2006) 2509–2518

DOI: 10.2140/agt.2006.6.2509

arXiv: math.GR/0606232

Abstract

Let $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Γ</mi></math>$ be a finitely generated, amenable group. Using an idea of É Ghys, we prove that if $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Γ</mi></math>$ has a nontrivial, orientation-preserving action on the real line, then $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Γ</mi></math>$ has an infinite, cyclic quotient. (The converse is obvious.) This implies that if $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Γ</mi></math>$ has a faithful action on the circle, then some finite-index subgroup of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Γ</mi></math>$ 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 2000

Primary: 20F60

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

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Publication

Received: 9 June 2006

Accepted: 1 September 2006

Published: 15 December 2006

Authors

Dave Witte Morris
Department of Mathematics and Computer Science University of Lethbridge Lethbridge Alberta, T1K 3M4 Canada
http://people.uleth.ca/~dave.morris/

Volume 6, issue 5 (2006)