## Geometry & Topology

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

#### Abstract

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 $Cr−1$ locally rigid.

#### Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513883419

Digital Object Identifier
doi:10.2140/gt.2004.8.877

Mathematical Reviews number (MathSciNet)
MR2087072

Zentralblatt MATH identifier
1079.57016

#### Citation

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. https://projecteuclid.org/euclid.gt/1513883419

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