Volume 6, issue 5 (2006)

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Amenable groups that act on the line

Dave Witte Morris

Algebraic & Geometric Topology 6 (2006) 2509–2518
 arXiv: math.GR/0606232
Abstract

Let $\Gamma$ be a finitely generated, amenable group. Using an idea of É 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.

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