Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 6, Number 5 (2006), 2509-2518.
Amenable groups that act on the line
Let be a finitely generated, amenable group. Using an idea of É Ghys, we prove that if has a nontrivial, orientation-preserving action on the real line, then has an infinite, cyclic quotient. (The converse is obvious.) This implies that if has a faithful action on the circle, then some finite-index subgroup of 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.
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
Permanent link to this document
Digital Object Identifier
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
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