Geometry & Topology, Vol. 8 (2004) Paper no. 23, pages 877--924.

Global rigidity of solvable group actions on S^1

Lizzie Burslem, Amie Wilkinson

Abstract. In this paper we find all solvable subgroups of Diff^omega(S^1) and classify their actions. We also investigate the C^r local rigidity of actions of the solvable Baumslag-Solitar groups on the circle.
The investigation leads to two novel phenomena in the study of infinite group actions on compact manifolds. We exhibit a finitely generated group Gamma and a manifold M such that:
* Gamma has exactly countably infinitely many effective real-analytic actions on M, up to conjugacy in Diff^omega(M);
* every effective, real analytic action of Gamma on M is C^r locally rigid, for some r>=3, and for every such r, there are infinitely many nonconjugate, effective real-analytic actions of Gamma on M that are C^r locally rigid, but not C^(r-1) locally rigid.

Keywords. Group action, solvable group, rigidity, Diff^omega(S^1)

AMS subject classification. Primary: 58E40, 22F05. Secondary: 20F16, 57M60.

DOI: 10.2140/gt.2004.8.877

E-print: arXiv:math.DS/0310498

Submitted to GT on 26 January 2004. Paper accepted 28 May 2004. Paper published 5 June 2004.

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Lizzie Burslem, Amie Wilkinson
Department of Mathematics, University of Michigan
2074 East Hall, Ann Arbor, MI 48109-1109 USA
Department of Mathematics, Northwestern University
2033 Sheridan Road, Evanston, IL 60208-2730 USA

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