A finitely generated, locally indicable group with no faithful action by C1 diffeomorphisms of the interval

Navas, Andrés

doi:10.2140/gt.2010.14.573

Download this article
	For screen
	For printing

Recent Issues

The Journal

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1364-0380 (online)

ISSN 1465-3060 (print)

Author Index

To Appear

Other MSP Journals

A finitely generated, locally indicable group with no faithful action by $C^1$ diffeomorphisms of the interval

Andrés Navas

Geometry & Topology 14 (2010) 573–584

DOI: 10.2140/gt.2010.14.573

Abstract

According to Thurston’s stability theorem, every group of $C^{1}$ diffeomorphisms of the closed interval is locally indicable (that is, every finitely generated subgroup factors through $ℤ$ ). We show that, even for finitely generated groups, the converse of this statement is not true. More precisely, we show that the group $F_{2} ⋉ ℤ^{2}$ , although locally indicable, does not embed into ${Diff}_{+}^{1} ((0, 1))$ . (Here $F_{2}$ is any free subgroup of $SL (2, ℤ)$ , and its action on $ℤ^{2}$ is the linear one.) Moreover, we show that for every non-solvable subgroup $G$ of $SL (2, ℤ)$ , the group $G ⋉ ℤ^{2}$ does not embed into ${Diff}_{+}^{1} (S^{1})$ .

Keywords

Thurston's stability, locally indicable group

Mathematical Subject Classification 2000

Primary: 20B27, 37C85, 37E05

References

Publication

Received: 25 February 2009

Revised: 22 October 2009

Accepted: 19 October 2009

Published: 26 January 2010

Proposed: Danny Calegari

Seconded: Leonid Polterovich and Yasha Eliashberg

Authors

Andrés Navas
Departamento de Matemáticas Facultad de Ciencia Universidad de Santiago de Chile Alameda 3363 Santiago Chile

Volume 14, issue 1 (2010)