Volume 8, issue 2 (2004)

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

Lizzie Burslem and Amie Wilkinson

Geometry & Topology 8 (2004) 877–924

arXiv: math.DS/0310498


In this paper we find all solvable subgroups of Diffω(S1) and classify their actions. We also investigate the Cr 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 Diffω(M);

(ii) every effective, real analytic action of Γ on M is Cr locally rigid, for some r 3, and for every such r, there are infinitely many nonconjugate, effective real-analytic actions of Γ on M that are Cr locally rigid, but not Cr1 locally rigid.

group action, solvable group, rigidity, $\mathrm{Diff}^{\omega}(S^1)$
Mathematical Subject Classification 2000
Primary: 58E40, 22F05
Secondary: 20F16, 57M60
Forward citations
Received: 26 January 2004
Accepted: 28 May 2004
Published: 5 June 2004
Proposed: Robion Kirby
Seconded: Martin Bridson, Steven Ferry
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
Illinois 60208-2730 USA