C1–actions of Baumslag–Solitar groups on S1

Guelman, Nancy; Liousse, Isabelle

doi:10.2140/agt.2011.11.1701

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$C^1$–actions of Baumslag–Solitar groups on $S^1$

Nancy Guelman and Isabelle Liousse

Algebraic & Geometric Topology 11 (2011) 1701–1707

DOI: 10.2140/agt.2011.11.1701

Abstract

Let $BS (1, n) = 〈 a, b ∣ a b a^{- 1} = b^{n} 〉$ be the solvable Baumslag–Solitar group, where $n \geq 2$ . It is known that $BS (1, n)$ is isomorphic to the group generated by the two affine maps of the line: $f_{0} (x) = x + 1$ and $h_{0} (x) = n x$ . The action on $S^{1} = ℝ \cup \infty$ generated by these two affine maps $f_{0}$ and $h_{0}$ is called the standard affine one. We prove that any faithful representation of $BS (1, n)$ into ${Diff}^{1} (S^{1})$ is semiconjugated (up to a finite index subgroup) to the standard affine action.

Keywords

circle diffeomorphism, solvable Baumslag–Solitar group

Mathematical Subject Classification 2010

Primary: 37C85

Secondary: 57S25, 37E10

References

Publication

Received: 28 October 2010

Revised: 6 April 2011

Accepted: 9 April 2011

Published: 3 June 2011

Authors

Nancy Guelman
IMERL Facultad de Ingeniería Universidad de la República Julio Herrera y Reissig 565 11300 Montevideo Uruguay

Isabelle Liousse
UFR de Mathématiques UMR CNRS 8524, Université de Lille 1 Laboratoire Paul Painlevé 59655 Villeneuve d’Ascq France

Volume 11, issue 3 (2011)