#### 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.

Notes on file formats
Lizzie Burslem, Amie Wilkinson

Department of Mathematics, University of Michigan

2074 East Hall, Ann Arbor, MI 48109-1109 USA

and

Department of Mathematics, Northwestern University

2033 Sheridan Road, Evanston, IL 60208-2730 USA

Email: burslem@umich.edu, wilkinso@math.northwestern.edu

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