## Algebraic & Geometric Topology

### Amenable groups that act on the line

Dave Witte Morris

#### 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.

#### Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796645

Digital Object Identifier
doi:10.2140/agt.2006.6.2509

Mathematical Reviews number (MathSciNet)
MR2286034

Zentralblatt MATH identifier
1185.20042

#### Citation

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. https://projecteuclid.org/euclid.agt/1513796645

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