Geometry & Topology

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

Lizzie Burslem and Amie Wilkinson

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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 r3, 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.

Article information

Geom. Topol., Volume 8, Number 2 (2004), 877-924.

Received: 26 January 2004
Accepted: 28 May 2004
First available in Project Euclid: 21 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58E40: Group actions 22F05: General theory of group and pseudogroup actions {For topological properties of spaces with an action, see 57S20}
Secondary: 20F16: Solvable groups, supersolvable groups [See also 20D10] 57M60: Group actions in low dimensions

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


Burslem, Lizzie; Wilkinson, Amie. Global rigidity of solvable group actions on $S^1$. Geom. Topol. 8 (2004), no. 2, 877--924. doi:10.2140/gt.2004.8.877.

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