Amenable groups that act on the line

Details Amenable groups that act on the line Amenable groups that act on the line

2006-06-09

arxiv.org/abs/math/0606232v3

Algebr. Geom. Topol. 6 (2006) 2509-2518

Mathematics

Group Theory

This is the version published by Algebraic & Geometric Topology on 15 December 2006

Scientific paper

10.2140/agt.2006.6.2509

Let Gamma be a finitely generated, amenable group. Using an idea of E Ghys, we prove that if Gamma has a nontrivial, orientation-preserving action on the real line, then Gamma has an infinite, cyclic quotient. (The converse is obvious.) This implies that if Gamma has a faithful action on the circle, then some finite-index subgroup of Gamma 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.

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