Amenable groups that act on the line

By Dave Witte Morris. 2006

83 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4059e1572a13c020fd5e9205a7bf63725250d429

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In Geometry and Topology

1. Adam Clay
2. D. Fisher
3. D. Rolfsen
4. Andrés Navas
5. Danny Calegari
6. Misha Gromov
7. Assaf Naor
8. Yann Ollivier

In Mathematical Physics

1. Adam Clay
2. D. Fisher
3. D. Rolfsen
4. Andrés Navas
5. Danny Calegari
6. Misha Gromov
7. Assaf Naor
8. Yann Ollivier

Citations in the corpus

• Foliations and the Geometry of 3-Manifolds (2007, background): “(Witte-Morris [251]) Let G be a left-orderable amenable group.”
• Groups acting on manifolds: around the Zimmer program (2008, methodology): “t on the line. An interesting approach might be to study the induced action of Γ on the space of left orders on Γ, for ideas about this approach, we refer the reader to work of Navas and Witte Morris [170, 176]. Perhaps the following should also be a conjecture, but here we only ask it is a question”
• Ordered Groups and Topology (2015, background): “[72] Suppose that G is a countable left-orderable group.”
• GROUP ACTIONS ON 1-MANIFOLDS: A LIST OF VERY CONCRETE OPEN QUESTIONS (2017, methodology): “The conjugacy action was brilliantly used by Witte-Morris in [133] to settle a question of Linnell [86], which was priorly raised -in the language of foliations- by Thurston [128]: Every finitelygenerated, left-orderable, amenable group admits a nontrivial homomorphism into Z.”
• Entropy and isoperimetry for linear and non-linear group actions (2008, background): “¤ fidg) infinite subgroups admit epimorphisms to Z [22].”
• On the dynamics of (left) orderable groups (2007, background): “This is indeed the main idea behind the proof of Morris-Witte’s theorem [41]: see §3.”