Algebraic & Geometric Topology

Amenable groups that act on the line

Dave Witte Morris

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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.

Article information

Algebr. Geom. Topol., Volume 6, Number 5 (2006), 2509-2518.

Received: 9 June 2006
Accepted: 1 September 2006
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F60: Ordered groups [See mainly 06F15]
Secondary: 06F15: Ordered groups [See also 20F60] 37C85: Dynamics of group actions other than Z and R, and foliations [See mainly 22Fxx, and also 57R30, 57Sxx] 37E05: Maps of the interval (piecewise continuous, continuous, smooth) 37E10: Maps of the circle 43A07: Means on groups, semigroups, etc.; amenable groups 57S25: Groups acting on specific manifolds

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


Morris, Dave Witte. Amenable groups that act on the line. Algebr. Geom. Topol. 6 (2006), no. 5, 2509--2518. doi:10.2140/agt.2006.6.2509.

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