Global rigidity of solvable group actions on S1

Burslem, Lizzie; Wilkinson, Amie

doi:10.2140/gt.2004.8.877

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Global rigidity of solvable group actions on $S^1$

Lizzie Burslem and Amie Wilkinson

Geometry & Topology 8 (2004) 877–924

DOI: 10.2140/gt.2004.8.877

arXiv: math.DS/0310498

Abstract

In this paper we find all solvable subgroups of ${Diff}^{ω} (S^{1})$ 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 $Γ$ and a manifold $M$ such that

(i) $Γ$ has exactly countably infinitely many effective real-analytic actions on $M$ , up to conjugacy in ${Diff}^{ω} (M)$ ;

(ii) every effective, real analytic action of $Γ$ on $M$ is $C^{r}$ locally rigid, for some $r \geq 3$ , and for every such $r$ , there are infinitely many nonconjugate, effective real-analytic actions of $Γ$ on $M$ that are $C^{r}$ locally rigid, but not $C^{r - 1}$ locally rigid.

Keywords

group action, solvable group, rigidity, $\mathrm{Diff}^{\omega}(S^1)$

Mathematical Subject Classification 2000

Primary: 58E40, 22F05

Secondary: 20F16, 57M60

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Publication

Received: 26 January 2004

Accepted: 28 May 2004

Published: 5 June 2004

Proposed: Robion Kirby

Seconded: Martin Bridson, Steven Ferry

Authors

Lizzie Burslem
Department of Mathematics University of Michigan 2074 East Hall Ann Arbor Michigan 48109-1109 USA

Amie Wilkinson
Department of Mathematics Northwestern University 2033 Sheridan Road Evanston Illinois 60208-2730 USA

Volume 8, issue 2 (2004)